A   TEXT-BOOK 


OF 


GEODETIC   ASTRONOMY 


BY 

JOHN  F.  HAYFORD,  C.E., 

Associate  Member  American  Society  of  Civil  Engineers; 
Expert  Computer  and  Geodesist  U.  5.  Coast  and  Geodetic  Survey. 


FIRST    EDITION, 
FIRST    THOUSAND. 


NEW   YORK: 

JOHN    WILEY   &   SONS. 

LONDON  :   CHAPMAN    &    HALL,    LIMITED. 

1898. 


Copyright,  1898, 

BY 

JOHN  F.  HAYFORD. 


ROBERT    DRUMMOND,    PRINTER,    NEW    YORK. 


PREFACE. 

(  To  be  read  by  th:  student  as  well  as  the  teacher.} 

THE  purpose  of  this  book  is  to  furnish  a  text  which  is 
sufficiently  short  and  easy  to  be  mastered  by  the  student  of 
civil  engineering  in  a  single  college  term,  but  which  shall  give 
him  a  sufficiently  exact  and  extensive  knowledge  of  geodetic 
astronomy  to  serve  as  a  basis  for  practice  in  that  line  after 
graduation.  Though  the  book  has  been  prepared  primarily 
for  students,  the  author  has  endeavored  to  insert  such  sub- 
ject-matter, tables,  and  convenient  formulae  as  would  make 
it  of  value  as  a  manual  for  the  engineer  making  astronomical 
observations.  In  order  to  make  the  book  sufficiently  short  it 
has  been  necessary  to  omit  all  mathematical  processes  except 
those  actually  necessary  for  developing  the  working  formulae. 
And  as  the  object  of  the  work  is  to  teach  a  certain  limited 
branch  of  astronomy,  rather  than  to  teach  mathematics,  the 
simpler  and  special  means  of  deriving  the  working  formulae 
have  been  chosen,  in  every  case  in  which  there  was  a  chance 
for  choice,  instead  of  the  more  difficult  and  general  derivation 
that  would  naturally  be  chosen  by  the  mathematician. 

The  occasion  for  the  book  is  the  fact  that  in  the  course  of 
study  prescribed  for  students  of  civil  engineering  at  Cornell 
University  but  five  hours  per  week  for  one  term  can  be 
devoted  to  the  text-book  work  and  lectures  on  astronomy. 
Under  these  conditions  it  is  out  of  the  question  to  use 
Chauvenet's  standard  work.  Even  Doolittle's  Practical 


IV  PREFACE. 

Astronomy  contains  more  mathematics  than  a  student  can  be 
expected  to  master  thoroughly  in  that  period.  Of  various 
other  text-books  available  none  seem  to  fit  the  special  condi- 
tions. 

In  the  wording  of  the  book  it  is  tacitly  assumed  that  the 
observer  is  in  the  northern  hemisphere.  To  make  the  word- 
ing general  would  require  too  many  circumlocutions. 

It  is  assumed  that  the  student  has  a  knowledge  of  least 
squares.  If,  however,  he  has  not  such  knowledge,  it  will  not 
debar  him  from  following  nearly  every  part  of  the  text  except 
§§  107-113,  dealing  with  the  treatment  of  transit  time  obser- 
vations by  least  squares,  and  §§  154-157,  giving  the  process 
of  combining  the  results  for  latitude  with  a  zenith  telescope 
by  that  method.  If  he  reads  carefully  §§  283-285,  stating 
the  technical  meaning  of  the  phrase  "  probable  error,"  the 
statement  of  the  uncertainty  of  a  given  observation  in  terms 
of  the  probable  error,  or  the  statement  of  the  errors  to  be  ex- 
pected from  certain  sources  in  such  terms,  should  convey  to 
him  a  definite  meaning. 

Considerable  space  has  been  devoted  in  the  text  to  a  dis- 
cussion of  the  various  sources  of  error  in  each  kind  of  obser- 
vation treated.  Two  separate  considerations  seem  to  the 
author  to  justify  this.  One  is  that  the  special  value  of 
geodetic  astronomy  as  a  part  of  the  course  of  training  of  an 
engineer  depends  largely  upon  the  fact  that  in  studying  it 
he  is  brought  face  to  face  with  the  idea  that  instruments  are 
fallible,  and  that  therefore  their  indications  must  be  carefully 
scrutinized  and  interpreted;  and  that  if  the  best  results  are 
to  be  secured  from  them,  the  sources  of  the  various  minute 
errors  which  combined  constitute  the  errors  of  observation 
must  be  carefully  studied.  The  other  consideration  is  that 
an  observer's  success  in  securing  accurate  results  with 
moderate  effort  depends  to  a  considerable  extent  upon  his 


PREFACE.  V 

power  to  estimate  rightly  the  relative  importance  of  the 
various  errors  affecting  his  final  result. 

The  accuracy  of  a  man's  thoughts,  as  well  as  of  his  speech, 
when  dealing  with  a  given  subject  depends  largely  upon  the 
precision  of  his  understanding  of  the  special  vocabulary  of 
that  subject.  With  that  idea  in  view  the  finder  list  of 
definitions  given  in  §  312  has  been  prepared.  The  student 
who  is  not  sure  of  the  exact  meaning  of  a  word  may  turn  to 
this  list  and  so  find  the  exact  definition  quickly.  In  reading 
definitions  the  context  should  also  be  read.  When  a  word  is 
defined  in  the  text  it  is  printed  in  italics. 

The  effort  has  been  made  to  select  the  formulae  which  have 
been  found  in  practice  to  lead  to  accurate  and  rapid  computa- 
tions. They  have  been  gathered  at  the  end  of  the  volume 
for  convenient  reference,  and  adjacent  to  each  formula  will  be 
found  references  to  the  corresponding  portion  of  the  text,  so 
that  for  those  who  may  use  the  book  as  a  manual  the  list  of 
formulas  with  these  references  may  serve  as  an  index  or  finder 
for  the  text. 

In  the  five  principal  chapters  the  instrument  has  first  been 
described,  and  the  adjustments  given,  as  well  as  directions  for 
observing,  and  an  example  of  the  record.  The  derivation  of 
the  formulae,  the  computation,  etc.,  follow.  If  the  text-book 
work  and  the  practical  work  of  the  observatory  are  carried  on 
together  during  the  same  term  one  naturally  wishes  the 
students  to  become  familiar  with  the  instruments  and  their 
manipulation  as  soon  as  possible.  In  that  case  it  is  recom- 
mended that  the  first  portions  only  of  certain  chapters  be 
taken  and  the  later  portions  omitted  temporarily.  The  fol- 
lowing order  may  then  be  used:  §§  1-27,  37,  51-63,  83-91, 
134-146,  177-187,  201-203,  205  to  middle  of  210,  273-276, 
28-50,  64-82,  92-133,  147-176,  188-272,  277  to  the  end. 

During  the  preparation  of  this  volume  the  text-books  on 


VI  PREFACE. 

astronomy  written  by  Chauvenet,  Doolittle  and  Loomis  have 
been  freely  consulted,  as  well  as  various  reports  of  the  Coast 
and  Geodetic  Survey,  of  the  Northern  Boundary  Survey,  of 
the  U.  S.  Lake  Survey,  and  the  report  of  the  Mexican 
Boundary  Survey  of  1892-93.  Appendix  No.  14  to  the 
Coast  Survey  Report  for  1880,  which  is  written  by  Assistant 
C.  A.  Schott  and  is  used  as  a  manual  by  the  officers  of  that 
survey,  has  been  extensively  drawn  upon  as  the  best  exposi- 
tion of  good  field  methods  known  to  the  author.  Several 
tables  have  been  taken  from  that  source,  notably  the  table  of 
factors  for  the  reduction  of  transit  time  observations  given  in 
§  299.  The  Superintendent  of  the  Coast  Survey  has  very 
kindly  furnished  certain  data,  and  photographs  of  instru- 
ments. 

JOHN    F.   HAYFORD. 
WASHINGTON,  D.  C.,  April  23,  1898. 


CONTENTS. 


CHAPTER  PAGE 

I.  INTRODUCTORY i 

Apparent  Motions I 

The  Earth 3 

Precession  and  Nutation 5 

Planets,  Satellites,  Stars 6 

Diurnal  Motion 8 

Definitions 10 

Time 18 

Questions  and  Examples 27* 

II.  COMPUTATION  OF  RIGHT  ASCENSION  AND  DECLINATION 30 

Position  of  Sun  and  Planets 31 

Interpolation 31 

Position  of  Moon 39 

Position  of  Stars 39 

Aberration 41 

Mean  Places 44 

Proper  Motion 48 

Computation  of  Apparent  Places 50 

Questions  and  Examples 55 

III.  THE  SEXTANT 59 

Description  of  Sextant 59 

Adjustments 62 

Directions  for  Observing  for  Time 65 

Record  and  Computation 71 

Parallax 73 

Refraction 75 

Derivation  of  Formula 80 

Discussion  of  Errors 82 

Eccentricity 88 

Other  Uses  of  the  Sextant 90 

Questions  and  Examples 94 

vii 


Vlii  CONTENTS. 


IV.  THE  ASTRONOMICAL  TRANSIT 96 

Descriptions  of  Transits 96 

Adjustments 99 

Directions  for  Observing 103 

The  Chronograph 105 

Record  and  Computation. 107 

Reduction  to  Mean  Line 108 

Inclination  Correction 112 

Correction  for  Diurnal  Aberration 1 16 

Azimuth,  Collimation,  and  Rate  Corrections 117,  118,  119 

Computation  Without  Least  Squares 120 

Least  Square  Computation 126 

Unequal  Weights 131 

Auxiliary  Observations 133 

Value  of  the  Level 135 

Discussion  of  Errors  140 

Miscellaneous 144 

Questions  and  Examples 146 

V.  THE  ZENITH  TELESCOPE  AND  THE  DETERMINATION  OF  LATITUDE.  150 

Description  of  Zenith  Telescope 151 

Adjustments 153 

Observing  List 156 

Directions  for  Observing 160 

Derivation  of  Formula 163 

Computation 166 

Combination  of  Results 168 

Micrometer  Value 174 

Discussion  of  Errors 181 

Other  Methods  of  Determining  Latitude 187 

Questions  and  Examples 194 

VI.  AZIMUTH. 197 

Description  of  Instrument 197 

Adjustments 200 

Directions  for  Observing  with  Direction  Instrument 201 

Record — Direction  Instrument 205 

The  Circle  Reading 2of 

Level  Correction «  209 

Azimuth  Formula 211 

Curvature  Correction 213 

Correction  for  Diurnal  Aberration 216 

Computation — Direction  Instrument 218 

Method  of  Repetitions 220 


CONTENTS.                    .  IX 

CHAPTER  PACK 

Micrometric  Method 223 

Discussion  of  Errors 230 

Other  Instruments  and  Methods 233 

Questions  and  Examples 238 

VII.  LONGITUDE 241 

Telegraphic  Method — Apparatus  and  Observations 242 

Telegraphic  Method — Computation 244 

Telegraphic    Method — Discussion    of    Errors    and    Personal 

Equation 248 

Longitude  by  Transportation  of  Chronometers 253 

Observations  upon  the  Moon 262 

Observations  upon  Jupiter's  Satellites 267 

VIII.  MISCELLANEOUS 268 

Suggestions  about  Observing 268 

Suggestions  about  Computing 270 

Probable  Errors 272 

Variation  of  Latitude 274 

Economics  of  Observing 276 

TABLES ( 279 

NOTATION  AND  PRINCIPAL  WORKING   FORMUL/E 328 

LIST  OF  DEFINITIONS 343 

INDEX 345 

FIGURES 352 


GEODETIC  ASTRONOMY, 


CHAPTER    I. 
INTRODUCTORY. 

1.  THIS  book  is  limited  to  the  treatment  of  astronomy  as 
applied  to  surveying,   or  to  what  might  be  called  geodetic 
astronomy.     Only  such  matters  are  treated  as  are  pertinent 
to  this  particular  limited  branch  of  the  subject.      Moreover, 
the  subject  as  thus  limited  is  treated  from  the  point  of  view 
of  the  engineer  who  wishes  to  obtain  definite  results,  rather 
than  from   that   of  a  mathematician  more   interested  in  the 
processes  concerned  than  in  their  final  outcome. 

2.  The   bodies   considered   by   the   engineer   in   geodetic 
astronomy  are  the  stars;  the  Sun;  the  planets,  including  the 
Earth;  the  Moon,  the  Earth's  satellite;  and  to  a  very  limited 
extent    some    of    the    satellites   of    the   other  planets.     The 
engineer  from  his  standpoint  upon  the  surface  of  the  Earth 
sees  these  different  bodies  moving  about  within  the  range  of 
his  vision, — aided  by  a  telescope  if  necessary.    Their  apparent 
motions  in  the  sky  as  seen  by  him  are  quite  complicated. 
His  success  in  locating  and  orienting  himself  upon  the  Earth 


2  GEODETIC  ASTRONOMY.  §  2. 

by  observations  upon  these  heavenly  bodies — for  that  is  his 
particular  purpose  in  observing  them — depends  first  of  all 
upon  his  having  a  clear  and  accurate  conception  of  their 
apparent  motions,  and  then  upon  his  possession  of,  and 
ability  to  use  efficiently,  the  instruments  with  which  the 
observations  are  made.  Much  of  the  complexity  in  the 
apparent  movements  of  these  heavenly  bodies  is  due  to  the 
fact  that  the  observer  sees  them  not  from  a  fixed  station  in 
space,  but  from  a  standpoint  upon  one  of  the  planets, — the 
Earth,  which  is  moving  rapidly  through  space  with  a  motion 
which  is  in  itself  quite  complicated.  He  sees  then  in  the 
apparent  motion  of  each  heavenly  body  upon  which  he  gazes 
not  only  the  actual  motion  of  that  body,  but  also,  reflected 
back  upon  him,  so  to  speak,  he  sees  the  actual  motion  of  the 
seemingly  solid  and  immovable  earth  upon  which  he  stands. 
He  is  like  a  passenger  upon  a  train  at  night  who  looks  out 
upon  the  many  lights  of  a  town.  He  sees  the  lights  all 
apparently  in  motion.  In  one  case  the  apparent  motion  of  the 
particular  light  may  be  entirely  due  to  his  own  motion  with  the 
train  upon  which  he  is  riding,  the  light  itself  being  at  rest. 
In  another  case  the  light  may  be  upon  another  moving  train 
and  its  apparent  motion  will  then  be  due  to  the  actual  motion 
of  each  of  the  trains.  If  the  darkness  is  sufficient  to  conceal 
the  landscape,  he  may  be  at  a  loss  to  determine  what  portions 
of  the  apparent  motions  of  the  lights  are  due  to  his  own 
change  of  position  and  what  to  the  motions  of  the  lights 
themselves.  He  is  then  in  the  position  of  a  man  when  he 
first  begins  to  study  the  apparent  movements  of  the  heavenly 
bodies. 

Let  us  first  form  concrete  conceptions  as  to  the  actual 
motion  of  each  of  the  bodies  under  consideration,  including 
the  Earth  itself.  We  will  then  be  in  a  position  to  understand 
the  apparent  motions. 


§  4-  THE  EARTH.  3 

3.  Conceive  the  Sun  to  be  a  very  large  self-luminous  mass 
of  matter.      For  the  present  let  it  be  supposed  to  be  fixed  in 
space.       Around    this    central    Sun    revolve     eight    planets, 
namely,  in  order  of    their  distance  from  the  Sun,  Mercury, 
Venus,  Earth,  Mars,  Jupiter,  Saturn,  Uranus,  and  Neptune. 
All  these  planets  move  nearly,  but  not  exactly,  in  the  same 
plane  passing  through  the  Sun.      The  orbit,  or  path,  of  any 
one  of  them  in  its  own  orbital  plane  is  very  nearly  a  perfect 
ellipse  with  one  focus  at  the  Sun,  and  the  velocity  with  which 
the  planet  moves  varies  at  different  parts  of  its  orbit  in  such 
a  way  that  the  line  joining  the  planet  and  Sun  describes  equal 
areas  in  equal  times.     This  orbit  and  law  of  velocity  result 
from  the  fact  that  each  planet  is  pursuing  its  path  in  obedi- 
ence to  a  single  force,  gravity,  continually  directed  toward  a 
fixed  center,  the  Sun. 

4.  The  Earth  may  be  taken  as  a  representative  planet. 
It  is  the  most  important  of  the  planets  for  our  present  purpose. 
It  moves  about  the  Sun  in  an  elliptical  orbit  at  a  mean  dis- 
tance from  the  Sun,  in  round  numbers,  of  92  800000  miles.* 
Though  the  orbit  is  an  ellipse,  its  major  and  minor  axes  are 
so  nearly  equal  that  if  it  were  plotted  to  scale  the  unaided 
eye  could  not  distinguish  it  from  a  circle.     The  greatest  dis- 
tance of  the  Earth  from  the  Sun  exceeds  the  least  distance  by 
but  little   more  than  3$.     The  Sun  is  in  that  focus  of  the 
ellipse  to  which  the  Earth  is  nearest  during  the  winter  (of  the 
northern  hemisphere).     The  eccentricity  of  the  ellipse,  and 
therefore    the    difference    of    the    two    axes,    is    very  slowly 
decreasing.     The  plane  of  the  Earth's  orbit  is  not  absolutely 
fixed  in  direction  in  space.      It  changes  with  exceeding  slow- 
ness— so    slowly,    in    fact,    that    it    is    used    as    one    of    the 
astronomical  reference  planes.     Moreover,  the  position  of  the 

*  See  "  The  Solar  Parallax  and  its  Related  Constants,"  Harkness,  p.  140. 


4  GEODETIC  ASTRONOMY.  §  5- 

elliptical  orbit  in  the  plane  is  slowly  changing;  that  is,  one 
focus  necessarily  remains  at  the  Sun,  but  the  direction  of  the 
major  axis  of  the  ellipse  gradually  changes. 

Roughly  speaking,  the  Earth  makes  one  complete  circuit 
of  its  orbit  in  the  period  of  time  which  is  ordinarily  called  one 
year.  At  different  portions  of  the  orbit  its  linear  velocity 
varies  according  to  the  law  (common  to  all  the  planets)  that 
the  line  joining  it  and  the  Sun  describes  equal  areas  in  equal 
times.  Each  portion  of  the  nearly  circular  path  being  almost 
perpendicular  to  the  line  joining  the  Earth  and  Sun  at  that 
instant,  the  linear  velocity  is  nearly  inversely  proportional  to 
the  distance  of  the  Earth  from  the  Sun.  Evidently  the  angular 
velocity  varies  still  more  largely  than  the  linear,  since  the 
greatest  linear  velocity  comes  at  the  same  time  as  the  least 
distance  from  the  Sun,  and  vice  versa. 

At  the  same  time  that  the  Earth,  as  a  whole,  is  swinging 
along  in  its  orbit  it  is  rotating  uniformly  about  one  of  its  own 
diameters  as  an  axis.*  This  rotation  is  so  nearly  uniform  in 
rate  that  it  is  assumed  to  be  exactly  uniform  and  is  used  to 
furnish  our  standard  of  time.  Roughly  speaking,  the  interval 
of  time  required  for  one  rotation  of  the  Earth  on  its  axis  is 
what  is  called  one  day.  The  more  exact  statement  will  be 
made  later. 

5.  The  axis  of  rotation  of  the  Earth  points  at  present 
nearly  to  the  star  called  Polaris,  or  North  Star,  and  makes  an 
angle  of  about  66£°  with  the  plane  of  the  Earth's  orbit,  or 
23^°  with  the  perpendicular  to  that  plane.  The  direction  of 
this  axis  of  rotation  is  not  fixed  in  space,  but  changes  just  as 


*  The  diameter  about  which  the  rotation  takes  place  is,  however,  not 
strictly  fixed  with  respect  to  the  Earth, — is  not,  in  other  words,  always  the 
same  diameter, — but  varies  through  a  range  of  a  few  feet  only  on  the  sur- 
face of  the  Earth.  See  §§286-7.  For  present  purposes,  however,  it  will  be 
considered  as  fixed. 


§  5-  PRECESSION  AND    NUTATION.  5 

the  axis  of  a  rapidly  spinning  top  is  seen  to  wabble  about. 
This  change  is  quite  slow,  but  extends  through  a  large  range 
of  motion.  It  is  compounded  of  two  motions  called  respec- 
tively precession  and  nutation.  By  virtue  of  the  motion  called 
precession  the  axis  of  the  earth  tends  to  remain  at  an  angle  of 
about  23!-°  with  the  perpendicular  to  the  plane  of  the  Earth's 
orbit  (usually  known  as  the  plane  of  the  ecliptic],  and  to 
revolve  completely  around  it,  describing  a  cone  of  two  nappes 
with  an  angle  of  about  47°  (twice  23^°)  between  opposite 
elements.  The  time  required  to  make  one  such  complete 
revolution  is,  at  the  present  rate,  about  26000  years.*  The 
motion  of  the  Earth's  axis  called  nutation  is  compounded  of 
several  periodic  motions,  the  principal  one  of  which  is  such 
as  to  cause  the  axis  to  describe  a  cone  of  which  the  right  sec- 
tion is  an  ellipse,  and  of  which  the  greatest  angle  between 
opposite  elements  is  about  eighteen  seconds  of  arc  and  the 
least  about  fourteen. f 

*  The  change  of  seasons  is  caused  by  the  inclination  of  the  Earth's  axis 
to  the  plane  of  its  orbit.  At  present  the  northern  end  of  the  axis  is  in- 
clined directly  away  from  the  Sun  at  about  Dec.  2ist;  the  Sun  then  appears 
to  be  farther  south  than  at  any  other  time,  and  it  is  winter  in  the  northern 
and  summer  in  the  southern  hemisphere.  At  about  June  2Oth  the  reverse 
is  true,  namely,  it  is  summer  in  the  northern  and  winter  in  the  southern 
hemisphere.  On  account  of  the  precession  the  winter  of  the  northern 
hemisphere  will  occur  in  June,  July,  and  August  about  13000  years  hence. 

f  All  the  various  motions  of  the  planets  and  their  satellites — the 
peculiar  mathematical  properties  of  their  orbits,  the  variability  of  the 
planes  of  the  orbits  and  of  the  orbits  in  the  planes,  etc. — are  by  celestial 
mechanics  shown  to  be  due  simply  to  the  action  of  gravitation.  Or,  stating 
the  matter  from  the  converse  point  of  view,  given  these  various  bodies  in 
their  actual  positions  and  having  their  actual  motions  at  a  given  instant, 
and  given  the  law  that  gravitation  acts  between  each  pair  of  them  with  an 
intensity  inversely  proportional  to  the  square  of  their  distance  apart  and 
directly  proportional  to  the  product  of  the  two  masses,  the  position  and 
motion  of  each  one  of  them,  their  orbits,  etc.,  at  any  other  stated  time  may 
be  computed  from  the  principles  of  celestial  mechanics  alone.  Even  the 
precession  and  nutation  are  caused  by  gravitation  and  are  thoroughly 


6  GEODETIC  ASTRONOMY.  §  6. 

6.  The  Earth  was  taken  as  representative  of  the  planets. 
Most  or  all  of  the  various  phenomena  which  have  been  indi- 
cated in  the  motion  of  the  Earth  are  repeated  in  each  of  the 
other  planets.  Each  has  an  orbital  plane  of  its  own  which  is 
slightly  variable  and  does  not  in  any  case  at  present  make  an 
angle  of  more  than  about  7°  with  the  plane  of  the  Earth's 
orbit.  Each  moves  in  that  plane  in  an  ellipse  of  which  the 
eccentricity  and  position  are  slowly  changing.  Each  has  a 
rotation  about  its  own  axis, — with  which,  however,  the  en- 
gineer is  not  concerned. 

The  Moon  is  a  satellite  of  the  Earth,  revolving  about  it 
under  the  action  of  gravity  just  as  the  Earth  revolves  about 
the  Sun,  and  in  its  orbit  are  again  found  the  same  peculiarities 
as  in  the  orbit  of  the  Earth  itself.  The  orbit  of  the  Moon  is 
an  ellipse  of  variable  eccentricity  and  position,  with  the  Earth 
in  one  focus,  and  lying  in  a  variable  plane  making  an  angle  of 
about  5°  with  the  plane  of  the  Earth's  orbit.  The  several 
variations  mentioned  are  much  greater  in  the  case  of  the 
Moon  than  of  the  Earth,  and  the  motion,  moreover,  is  subject 
to  other  perturbations.  Its  motion  is  therefore  a  most  diffi- 
cult one  to  compute. 

Each    one    of    the    other    planets,    except    Mercury    and 

accounted  for  by  principles  of  celestial  mechanics  derived  from  the  above 
law  of  gravitation.  The  luni-solar  precession  is  due  to  the  fact  that  the 
Earth  is  not  a  sphere,  but  a  spheroid,  having  an  excess  of  matter  in  the 
equatorial  regions.  One  component  of  the  attraction  of  the  Moon  and  Sun 
acting  upon  this  equatorial  excess  tends  continually  to  shift  the  position 
of  the  equator  in  one  direction  without  changing  its  angle  with  the  eclip- 
tic. The  action  of  the  planets  upon  the  Earth  as  a  whole  tends  to  draw  it 
out  of  the  plane  of  its  orbit,  or  rather  to  change  the  orbital  plane.  This 
change  is  called  the  planetary  precession.  The  luni-solar  and  planetary 
precessions  together  constitute  what  is  often  called  simply  the  precession. 
Nutation  is  made  up  of  periodic  motions  which  are  due  to  regular  periodic 
fluctuations  in  the  forces  which  produce  precession.  Nutation  might  be 
described  as  the  periodic  part  of  precession 


§  7-  THE   STARS.  7 

Venus,  has  one  or  more  satellites  bearing  the  same  relations 
to  it  that  the  Moon  does  to  the  Earth. 

The  comparatively  erratic  motions  of  the  numerous 
asteroids,  or  small  planets,  moving  in  orbits  between  that  of 
Mars  and  Jupiter,  and  of  the  comets  and  meteors  which 
occasionally  visit  the  solar  system,  prevent  their  use  by  the 
engineer.* 

The  Sun,  the  eight  planets  and  their  satellites,  and  the 
asteroids  together  constitute  the  solar  system. 

7.  The  stars  are  self-luminous  bodies  at  great  distances 
from  the  solar  system.  Their  remoteness  is  to  a  certain 
extent  indicated  by  the  fact  that  with  the  best  telescopes  and 
with  the  highest  magnifying  powers  at  present  available  the 
image  of  a  star  cannot  be  magnified.  It  remains  with  all 
powers  and  telescopes  a  point  of  light  of  which  the  apparent 
size  is  merely  a  measure  of  the  imperfection  of  the  telescope 
and  eye.  But  the  best  evidence  of  the  immense  distance  even 
to  the  nearest  of  the  fixed  stars  is  the  fact  that  even  though 
the  diameter  of  the  Earth's  orbit,  186  million  miles,  be  taken 
as  the  base  of  a  triangle  of  which  the  vertex  is  at  the  star,  it 
is  only  with  the  greatest  difficulty,  if  at  all,  that  the  angle  at 
the  star  can  be  detected  even  though  the  instruments  used  be 
of  the  highest  order  of  accuracy  and  a  long  series  of  observa- 
tions are  used.  In  the  few  cases  in  which  this  angle  at  the 
star  has  been  successfully  measured  it  has  been  found  to  be 
not  greater  than  one  second  of  arc.  For  the  purposes  of  the 
engineer,  then,  it  may  be  assumed  that  each  and  every  star  is 
at  so  great  a  distance  from  the  Earth  that  the  true  direction 
in  space  of  the  straight  line  from  the  Earth  to  the  star  is  the 
same  at  all  times  of  the  year  notwithstanding  the  widely 
separated  positions  the  Earth  may  occupy  in  its  orbit. 

*  For  an  interesting  treatment  of  comets,  meteors,  and  asteroids,  see 
Young's  General  Astronomy,  and  Chamber's  Astronomy,  pp.  104-109, 
278-430,  780-816. 


8  GEODETIC  ASTRONOMY.  §  8. 

If  the  stars  had  no  motion  relative  to  each  other  or  to  the 
solar  system  as  a  whole,  the  true  direction  of  the  line  from 
the  Earth  to  any  one  star  would  not  vary  from  year  to  year. 
As  a  matter  of  observation,  however,  it  is  known  that  in 
general  the  true  direction  of  such  a  line  does  change,  although 
the  change  is  exceedingly  slow  in  every  case.  This  change 
will  be  treated  more  in  detail  in  a  later  chapter. 

The  apparent  motion  of  any  particular  heavenly  body  as 
seen  by  an  observer  upon  the  Earth  is  the  compound  result 
of  the  motion  of  the  Earth  and  of  that  body. 

8.  In  the  case  of  a  star,  the  object  observed  is  for  most 
purposes  at  what  may  be  considered  an  infinite  distance. 
The  line  joining  the  observer  and  star  preserves,  therefore,  a 
sensibly  constant  direction  in  spite  of  the  motion  through 
space  of  the  observer  upon  the  Earth.  The  apparent  motion 
of  the  star  is  caused  by  the  rotation  of  the  Earth  about  its 
axis  and  the  change  in  the  direction  of  that  axis  in  space. 
The  rotation  of  the  Earth  causes  the  line  of  sight  to  a  star  to 
seem  to  describe  at  a  uniform  rate  a  right  circular  cone,  of 
which  the  axis  is  the  line  joining  the  observer  with  a  point  in 
the  sky  at  an  infinite  distance  in  the  axis  of  the  Earth  pro- 
duced. In  other  words,  the  axis  of  the  cone  is  a  line  from 
the  observer  parallel  to  the  axis  of  the  Earth.  Such  a  line, 
for  any  point  in  the  northern  hemisphere,  pierces  the  sky  in 
a  point  not  far  from  the  North  Star,  Polaris.  The  angle 
between  any  element  of  the  cone  and  its  axis  is  the  angle 
between  the  line  joining  observer  to  star,  and  the  axis  of  the 
Earth.  This  angle  is  called  the  polar  distance  of  the  star, — 
north  polar  distance  if  measured  from  the  north  end  of  the 
Earth's  axis.  So  long  as  these  two  lines  are  fixed  in  direction 
in  space,  the  line  of  sight  to  the  star  continues  to  describe 
the  same  right  circular  cone  once  for  every  turn  which  the 
Earth  makes  on  its  axis.  For  example,  the  line  of  sight  to 


§  8.          APPARENT  DIURNAL   MOTION  OF   THE   STARS.         9 

Polaris  makes  an  angle  of  about  i  J°  with  the  axis  of  the  Earth, 
and  describes  a  corresponding  right  circular  cone.  Or  it  may 
be  said  that  Polaris  seems  to  describe  a  circle  in  the  sky  of 
which  the  radius  subtends  an  angle  at  the  eye  of  ij°.  With 
a  good  telescope  Polaris  may  be  followed  completely  around 
the  circle,  all  of  which  would  be  above  the  horizon  for  any 
point  in  the  United  States.  With  the  naked  eye  only  that 
portion  of  the  apparent  motion  which  occurs  during  the 
hours  of  darkness  could  be  observed.  For  an  observer  at 
Ithaca,  in  latitude  42^°,  the  cone  for  a  star  having  a  north 
polar  distance  less  than  42^°  is  entirely  above  the  horizon. 
Given  one  view  of  the  star  and  an  idea  of  the  position  of  the 
vanishing  point  of  the  Earth's  axis  in  the  sky,  an  observer 
is  able  to  trace  out  the  whole  apparent  path  of  the  star.  For 
Ithaca,  a  star  of  north  polar  distance  of  42^°  has  its  cone 
tangent  to  the  horizon;  and  if  greater  than  that  value,  a  part 
of  the  cone  must  be  below  the  horizon,  and  the  star  is  neces- 
sarily invisible  on  that  portion.  If  the  north  polar  distance 
is  90°,  the  cone  becomes  a  plane.  Stars  still  farther  south 
describe  a  right  circular  cone  about  the  southern  portion  of 
the  Earth's  axis  produced,  the  angle  of  the  cone  being  the 
south  polar  distance  of  the  star. 

In  every  case  the  diurnal  rotation  of  the  Earth  causes  the 
line  of  sight  to  a  star  to  describe  a  right  circular  cone.  But, 
as  has  already  been  stated  (§  5),  the  direction  of  the  Earth's 
axis  is  continually  changing  slowly,  and  hence  the  north  polar 
distance  or  angle  between  the  Earth's  axis  and  the  line  join- 
ing the  observer  and  star  is  continually  changing.  The  cone 
of  revolution  therefore  slowly  changes  from  day  to  day. 

If  the  object  observed  is  not  a  star  at  a  practically  infinite 
distance,  but  a  planet,  the  Moon,  or  the  Sun,  at  a  finite  dis- 
tance, the  line  joining  observer  to  object  describes  a  surface 
any  small  portion  of  which  may  be  considered  to  be  a  portion 


IO  GEODETIC  ASTRONOMY.  §  IO. 

of  the  surface  of  a  right  circular  cone.  But  the  north  polar 
distance  of  the  object  now  continually  changes,  not  only  on 
account  of  the  change  in  the  direction  of  the  Earth's  axis,  but 
still  more  largely  on  account  of  the  change  in  the  true  direc- 
tion of  the  line  joining  observer  and  object, — two  points 
which  are  at  a  finite  distance  from  each  other  and  both  in 
motion.  This  last  cause  also  makes  the  rate  at  which  the 
surface  is  described  variable. 

9.  The  two  principal  reference  planes  of  astronomy  are 
the  plane  of  the  equator  and  the  plane  of  the  Earth's  orbit, 
or,  as  it  is  generally  called,  \.\\.z  plane  of  ecliptic.     The  plane 
of  the  equator  is  a  plane  passing  through  the   center  of  the 
Earth  and  perpendicular  to  its  axis  of  rotation.      Neither  of 
these  two  planes,  from  what  has  already  been  written,   are 
fixed   in   space,    nor  fixed   relatively  to   each   other.     Their 
changes  of  position  are,  however,  very  slow. 

10.  To  avoid  the  necessity  of  using  cumbersome  expres- 
sions and  circumlocutions,  it  is  convenient  to  make  use  of  the 
celestial  sphere  as  an  arbitrary  conception.     The  celestial  sphere 
is  a  sphere  of  infinite  radius,  the  eye  of  the  observer  being 
supposed  to  be  at  its   center.     Any  celestial  object  is  consid- 
ered to  be  projected  along  the  line  of  sight  to  the  surface  of 
this  sphere  and  is  referred  to  as  occupying  that  position  upon 
the  sphere.     Then  for  convenience  one  may  speak  of  arcs, 
angles,  and  triangles  upon  the  celestial  sphere  instead  of  using 
the  complicated  expressions  necessary  in  speaking  always  of 
the  actual  lines  and   planes  which  are  under  consideration. 
The  sphere  is  assumed  to  be  of  infinite  radius  so  that  lines 
which  are  parallel  and  at  a  finite  distance  apart  will  intersect 
the  sphere  in  the  same  point,  or  at  least  what  is  sensibly  one 
point,  since  two  points  at  a  finite  distance  apart  must  appear 
as  one  when  seen  from  an  infinite  distance.     So  also  parallel 
planes   which    are    at    a    finite   distance  apart  intersect    the 


§  12.  DEFINITIONS.  II 

celestial  sphere  in  the  same  arc.  For  example,  the  axis  of 
the  Earth  and  a  line  parallel  to  it  through  the  eye  of  the 
observer  both  intersect  the  celestial  sphere  in  the  same  pair 
of  points  called  the  poles  of  the  equator,  or  more  briefly  the 
poles, — north  and  south  respectively.  Also  the  plane  of  the 
equator,  and  a  plane  parallel  to  it  through  the  eye  of  the 
observer,  intersect  the  celestial  sphere  in  the  same  great  circle 
which  is  called  the  equator  of  the  celestial  sphere,  or  more  fre- 
quently simply  the  equator. 

11.  The  equator,  the  ecliptic,  hour-circles,  and  the  horizon 
are   all   great   circles  of   the  celestial  sphere   formed  by  the 
intersection  of  various  planes  with  that  sphere. 

The  ecliptic  is  the  intersection  of  the  plane  of  the  ecliptic, 
or,  in  other  words,  the  plane  of  the  Earth's  orbit,  with  the 
celestial  sphere.  The  Sun,  therefore,  is  always  seen  pro- 
jected on  some  point  of  the  ecliptic. 

An  hour-circle  is  the  intersection  of  a  plane  passing 
through  the  Earth's  axis  with  the  celestial  sphere.  All  hour- 
circles  are  then  great  circles  passing  through  the  poles. 

The  horizon  is  the  intersection  with  the  celestial  sphere  of 
a  plane  passed  through  the  eye  of  the  observer  perpendicular 
to  the  plumb-line,  or  line  of  action  of  gravity,  at  the  observer. 
All  horizontal  lines  at  a  given  point  on  the  Earth's  surface 
pierce  the  celestial  sphere  in  the  horizon  of  that  point. 

In  each  of  these  cases  it  is  evident  that  the  great  circle  on 
the  celestial  sphere  would  not  be  changed  if  the  intersecting 
plane  were  moved  parallel  to  itself  a  finite  distance, — for 
instance,  to  pass  through  any  other  point  in  or  upon  the  sur- 
face of  the  Earth.  For  example,  the  horizon  may  be  consid- 
ered to  be  the  intersection  with  the  celestial  sphere  of  a  plane 
passing  through  the  center  of  the  Earth  and  perpendicular  to 
the  observer's  gravity  line,  instead  of  that  given  above. 

12.  The   angle   between   a  line  joining  the  center  of  the 


12  GEODETIC  ASTRONOMY.  §  12. 

Earth  to  a  star  (or  other  celestial  object)  and  the  plane  of  the 
equator  is  called  the  declination  of  that  object.  It  is  meas- 
ured upon  the  celestial  sphere  by  that  portion  of  the  hour- 
circle  passing  through  the  star  which  is  between  the  star  and 
the  equator.  The  declination  is  considered  positive  when 
measured  north  from  the  equator.  It  follows  from  the 
definition  of  polar  distance  (given  in  §  8)  that  the  declination 
and  polar  distance  are  complements  of  each  other. 

The  equator  and  the  ecliptic  intersect  each  other  at  an 
angle  of  about  23°  27'.  Their  two  points  of  intersection  on 
the  celestial  sphere  are  called  the  equinoxes.  That  one  at 
which  the  Sun  is  found  in  the  spring  is  called  the  vernal 
equinox,  and  that  at  which  it  is  found  in  the  fall  the  autumnal 
equinox.  As  both  the  equator  and  ecliptic  move  slowly  in 
space  the  equinoctial  points  slowly  shift  in  position  upon  the 
celestial  sphere. 

The  right  ascension  of  a  star,  or  other  celestial  object,  is 
the  angle,  measured  along  the  equator,  between  the  two 
hour-circles  which  pass  through  the  star  and  the  vernal  equi- 
nox respectively.  In  other  words,  the  righ't  ascension  is  the 
angle  between  two  planes,  one  passing  through  the  Earth's 
axis  and  the  star  and  the  other  through  the  Earth's  axis  and 
the  vernal  equinox.  It  is  reckoned  in  degrees  from  o  to  360, 
in  the  direction  that  would  appear  counter-clockwise  if  one 
looked  toward  the  equator  from  the  north  pole, — from  west 
to  east.  Right  ascensions  are  still  more  frequently  expressed 
in  time,  24  hours  being  equivalent  to  360  degrees. 

The  zenith  is  the  point  in  which  the  action-line  of  gravity 
produced  upward  intersects  the  celestial  sphere.  The  oppo- 
site point  on  the  celestial  sphere  is  called  the  nadir. 

The  intersection  with  the  celestial  sphere  of  a  plane  passed 
through  its  center,  the  zenith,  and  the  pole  is  called  the 
meridian,  and  the  plane  itself  is  called  the  meridian  plane* 


§  13-  DEFINITIONS.  13 

The  intersection  of  the  meridian  plane  with  the  plane  of  the 
horizon  is  called  the  meridian  line.  It  connects  the  north 
and  south  points  of  the  horizon.  The  intersection  with  the 
celestial  sphere  of  a  plane  through  the  zenith  perpendicular 
to  the  meridian  plane  is  called  the  prime  vertical.  The  east 
and  west  points  of  the  horizon  are  in  the  prime  vertical. 

13.  The  angle,  measured  along  the  equator,  between  the 
meridian  and  the  hour-circle  passing  through  a  star  (or  other 
celestial  object)  is  the  hour -angle  of  the  star.  In  other  words, 
the  hour-angle  is  the  angle  between  the  meridian  plane  and  a 
plane  passing  through  the  Earth's  axis  and  the  star.  Hour- 
angles  are  reckoned  like  right  ascensions,  either  in  degrees, 
minutes,  and  seconds  of  arc  or  in  hours,  minutes,  and  seconds 
of  time.  In  this  book  hour-angles  will  be  measured  for  180° 
each  way  from  the  upper  branch  of  the  meridian  and  will 
always  be  considered  positive. 

The  student  should  distinguish  carefully  between  an  hour- 
angle  and  a  right  ascension.  Each  is  an  angle  between  two 
planes.  In  each  case  one  of  the  two  planes  is  defined  by  the 
Earth's  axis  and  the  star,  and  therefore  changes  direction  but 
slowly  in  space.  The  second  plane  concerned  in  the  case  of 
a  right  ascension  is  defined  by  the  Earth's  axis  and  the  vernal 
equinox.  This  plane  changes  its  direction  very  slowly.  So 
the  right  ascension  of  a  star  is  an  angle  which  is  slowly  chang- 
ing,— at  a  rate  of  less  than  one  minute  of  arc  per  year  for 
nearly  all  the  stars.  The  second  plane  concerned  in  the 
measurement  of  an  hour-angle  is  the  plane  of  the  meridian. 
This  accompanies  the  Earth  in  its  diurnal  rotation.  Hence 
the  hour-angle  of  a  celestial  object  varies  rapidly, — 360°  for 
each  rotation  of  the  Earth  on  its  axis.  The  right  ascension 
and  declination  are  spherical  co-ordinates  locating  a  celestial 
object  with  reference  to  the  hour-circle  through  the  vernal 
equinox  and  the  equator.  The  hour-angle  and  declination 


14  GEODETIC  ASTRONOMY.  §  1 5. 

are  two  spherical  co-ordinates  locating  a  celestial  object  with 
reference  to  the  meridian  and  equator. 

14.  It  is  convenient  for  some  purposes  to  refer  the  posi- 
tion  of   a  heavenly  body  by   spherical   co-ordinates   to    the 
planes  of  the  meridian  and  horizon, — the  two  co-ordinates  in 
this  case  being  the  altitude  and  azimuth.      The  altitude  of  a 
heavenly  body  is  its  angular  distance  above  the  horizon,  or 
the  angle  between  the  line  joining  the  observer  to  the  star, 
and  the  horizontal  plane.     Any  great  circle  of  the  celestial 
sphere  passing  through  the  zenith  is  called  a  vertical  circle. 
The  altitude  of  a  star  is  measured  by  that  portion  of  the  verti- 
cal circle  passing  through  the  star  which  is  included  between 
the  star  and  the  horizon.     The  azimuth  of  a  star,  or  other 
celestial  body,  is  the  angle  between  the  plane  of  the  meridian 
and  the  vertical  plane  passing  through  the  star.     The  same 
definition  applies  to  a  line  joining  two  terrestrial  points.     The 
azimuth  at  station  A  on  the  Earth's  surface,  of  the  line  join- 
ing stations  A  and  B,  is  the  angle  between  the  vertical  plane 
at  A   passing  through  the  line  AB  and  the  meridian  plane 
of  A.     The  azimuth  of  a  star  is  measured  on  the  celestial 
sphere  by  that  portion  of  the  horizon  included  between  the 
star's  vertical  circle  and  the  meridian  line.      In  general  the 
altitude  and  azimuth  of  a  celestial  object  are  both  changing 
rapidly  because  of  the  Earth's  rotation.     The  zenith  distance 
of  a  star  is  its  angular  distance  from  the  zenith, — measured, 
of  course,  along  a  vertical  circle.     The  zenith  distance  and 
altitude  are  complements  of  each  other. 

15.  The  astronomical  latitude  of  a  station  on  the  surface 
of  the  Earth  is  the  angle  between  the  line  of  action  of  gravity 
at  that  station  and  the  plane  of  the  equator.      It  is  measured 
on  the  celestial  sphere  along  the  meridian  from  the  equator 
to  the  zenith. 

The  astronomical  longitude  of  a  station  on  the  surface  of 


§15-  LATITUDE  AND    LONGITUDE.  1 5 

the  Earth  is  the  angle  between  the  meridian  plane  of  that 
station  and  some  arbitrarily  chosen  initial  meridian  plane. 
Usually  the  meridian  of  Greenwich,  Eng.,  is  taken  as  the 
initial  meridian,  but  sometimes  that  of  Paris  or  of  Berlin,  or 
in  the  case  of  detached  surveys  some  arbitrary  meridian  plane 
to  which  all  points  of  the  survey  may  be  conveniently  referred. 
Unless  otherwise  stated  astronomical  latitude  or  astronomical 
longitude  is  meant  when  the  word  latitude  or  longitude  is 
used  in  this  book. 

The  student  should  distinguish  astronomical  latitude  and 
longitude  from  geodetic  latitude  and  longitude,  and  should  be 
careful  not  to  confuse  either  one  of  these  with  celestial  latitude 
and  longitude.  The  geodetic  latitudes  and  longitudes  differ 
from  the  astronomical  in  that,  instead  of  being  referred  to  the 
actual  action-line  of  gravity  at  the  station,  they  are  referred 
to  a  gravity  line  which  has  been  corrected  for  local  deflection, 
or  station  error.*  Celestial  latitudes  and  longitudes  form  a 

*  In  the  operations  of  geodesy  the  action-line  of  gravity  has  been 
found  to  be  nearly  perpendicular  at  all  stations  to  the  surface  of  an 
imaginary  ellipsoid  of  revolution  generated  by  the  revolution  of  an  ellipse 
about  its  minor  axis,  the  minor  axis  coinciding  with  the  axis  of  rotation  of 
the  Earth.  This  is  the  form  which  a  rotating  liquid  mass  necessarily 
assumes  under  the  action  of  no  other  forces  than  the  action  of  gravitation 
between  its  component  parts.  Values  for  the  polar  and  equatorial  diam- 
etersi  respectively,  of  this  ellipsoid  having  been  determined  such  that  its 
surface  is  as  nearly  as  possible  perpendicular  at  all  points  to  the  action- 
lines  of  gravity,  the  outstanding  difference  of  direction  between  the  normal 
to  the  surface  of  the  ellipsoid  at  any  point  and  the  actual  action-line  of 
gravity  at  that  point  is  called  the  station  error,  or  local  deflection  of  the 
vertical  at  that  point.  The  station  error  is  supposed  to  be  due  to  varia- 
tions of  density  in  the  interior  of  the  Earth  near  the  station,  and  to  the 
local  irregularities  of  the  surface. 

The  operation  of  determining  the  station  error  at  a  given  place  is  as 
follows:  The  astronomical  latitude  and  longitude  of  each  of  a  number  of 
stations  are  determined.  The  stations  are  connected  by  an  accurate 
geodetic  survey.  All  the  latitudes  and  longitudes  are  then  reduced  to  one 
of  the  stations  by  use  of  the  known  elements  of  the  ellipsoid.  The  mean 


1 6  GEODETIC  ASTRONOMY.  §  1 6. 

system  of  spherical  co-ordinates, — frequently  used  by  the 
astronomer  but  seldom  by  the  engineer.  In  this  the  ecliptic 
and  vernal  equinox  play  the  same  part  as  do  the  equator  and 
vernal  equinox  in  the  case  of  declinations  and  right  ascen- 
sions. 

16.  In  general,  when  the  engineer  observes  a  heavenly 
body  he  has  one  of  four  objects  in  view,  namely,  to  determine 
his  astronomical  latitude,  the  azimuth  of  a  line  joining  his 
station  with  some  other  terrestrial  point,  the  true  local  time 
at  the  instant  of  observation,  or  the  longitude  of  his  station. 
The  determination  of  longitude  always  involves  a  determina- 
tion of  the  true  local  time  together  with  additional  operations 
which  are  in  some  cases  quite  complicated.  The  instrument 
used  in  any  case  for  the  determination  of  time,  latitude,  or 
azimuth  indicates  the  position  of  the  horizon,  and  conse- 
quently of  the  zenith,  by  means  of  attached  spirit-levels,  or 

of  the  various  values  of  the  latitude  of  this  single  station  as  thus  obtained 
is  called  its  geodetic  latitude.  The  corresponding  statement  applies  to  the 
longitude.  It  is  evident  that  the  greater  the  number  of  stations  and  the 
more  widely  scattered  they  are  the  nearer  will  the  vertical  as  given  by  the 
geodetic  latitude  and  longitude  coincide  with  the  normal  to  the  ellipsoid. 
The  difference  between  the  astronomical  and  geodetic  latitude  at  a  given 
point  is  therefore  usually  called  the  station  error  in  latitude.  A  similar 
statement  defines  station  error  in  longitude.  For  further  information  on 
this  subject  see  Clark's  Geodesy,  pp.  287-288,  Merriman's  Geodetic  Sur- 
veying, pp.  79-88,  or  any  extended  treatise  on  geodesy. 

Station  errors  in  longitude,  or  deflections  of  the  vertical  at  right  angles 
to  the  meridian,  change  the  plane  of  the  meridian  from  the  position  it  would 
otherwise  occupy  and  so  change  all  azimuths  from  the  values  they  would 
otherwise  have.  Hence  there  arises  the  same  distinction  between  the 
astronomical  azimuth  of  a  line  and  its  geodetic  azimuth  as  is  drawn  above 
between  the  astronomical  and  geodetic  latitudes  and  longitudes.  On 
account  of  station  error  the  line  of  gravity  at  a  station  and  the  axis  of  the 
Earth  do  not,  in  general,  intersect.  Hence  to  be  exact  the  meridian  plane 
must  be  said  to  be  defined,  not  by  the  line  of  gravity  and  the  Earth's  axis 
of  rotation,  but  by  the  line  of  gravity  and  the  point  in  which  the  axis  of 
rotation  produced  intersects  the  celestial  sphere. 


§  1 6.  PURPOSE   OF  OBSERVATIONS.  1 7 

by  a  basin  of  mercury  having  a  free  horizontal  surface.  The 
star,  or  other  celestial  object,  is  usually  observed  with  a  tele- 
scope. The  two  points  on  the  celestial  sphere  always 
observed  are,  therefore,  the  zenith  and  the  object.  The  right 
ascension  and  declination  of  the  object  observed  become 
known,  independently  of  the  observations,  by  the  methods 
indicated  in  the  next  chapter.  The  process  most  frequently 
used  is  to  acquire,  by  instrumental  observation  and  by  the 
means  indicated  in  the  next  chapter,  a  knowledge  of  three  of 
the  elements,  arcs  and  angles,  of  some  triangle  on  the  celes- 
tial sphere  of  which  one  of  the  unknown  elements,  now 
capable  of  computation,  is  the  quantity  sought,  or  is  one 
from  which  the  required  quantity  can  be  readily  derived. 

For  example,  suppose  that  the  latitude  of  the  station  of 
observation  is  known,  and  that  the  zenith  distance  of  a  certain 
star  is  accurately  observed.  Let  the  true  local  sidereal  time 
(see  §  1 8)  at  the  instant  of  observation  be  required.  In  the 
triangle  on  the  celestial  sphere  defined  by  the  pole,  the  zenith, 
and  the  instantaneous  position  of  the  star,  the  arc  from  the 
zenith  to  the  pole  is  known,  being  the  complement  of  the 
latitude  of  the  station.  The  arc  from  the  star  to  the  pole 
becomes  known  by  the  methods  indicated  in  the  next  chapter, 
since  it  is  the  complement  of  the  declination  of  the  star  at 
the  instant  of  observation.  The  arc  from  the  zenith  to  the 
star,  the  zenith  distance,  was  directly  observed.  Hence  in  the 
spherical  triangle  pole-zenith-star  all  three  arcs  are  known  and 
any  of  the  angles  may  be  computed.  The  angle  at  the  pole 
of  that  triangle  is  the  hour-angle  of  the  star  at  the  instant  of 
observation.  This  being  computed  by  the  methods  of 
spherical  trigonometry,  a  mere  addition  to  or  subtraction 
from  the  right  ascension  of  the  star  (which  becomes  known 
by  the  methods  of  the  following  chapter)  gives  the  true  local 
sidereal  time  (as  will  .be  shown  later). 


1 8  GEODETIC  ASTRONOMY.  §  1 8. 

17.  On  account  of  the  rapid   apparent  motion   of  most 
celestial  objects,   time   enters  as  an  important  element  into 
almost  every  astronomical  problem  with  which  the  engineer 
has  to  deal.     Three  kinds  of  time  are  in  use  in  astronomy: 
sidereal  time,  apparent  solar  time,  and  mean  solar  time. 

The  passage  of  a  star  or  other  celestial  object  across  the 
meridian  is  called  its  transit  or  culmination. 

The  meridian  (a  great  circle  of  the  celestial  sphere)  is 
divided  into  two  half-circles  by  the  poles.  If  the  whole  of 
the  meridian  be  considered,  a  star  has  two  transits  for  each 
complete  rotation  of  the  Earth  on  its  axis :  one  over  that  half 
of  the  meridian  stretching  from  pole  to  pole  which  includes 
the  zenith,  and  the  other  over  that  half  which  passes  through 
the  nadir.  The  first  of  these  is  called  the  upper  transit  or 
upper  culmination,  and  the  second  the  lower  transit  or  lower 
culmination.  The  word  transit  or  culmination  unmodified 
usually  means  the  upper  transit.  The  expression  "  the 
passage  of  a  star  across  the  meridian  "  refers,  of  course,  to  the 
apparent  motion  of  the  star.  It  would  be  more  accurate  to 
say  that  the  meridian  passes  the  star.  But  to  refer  directly 
to  the  apparent  motion  as  if  it  were  real  saves  circumlocution, 
is  more  clear  in  many  cases,  and  is  not  misleading  if  one  keeps 
in  mind  that  this  is  merely  a  mode  of  speech. 

18.  A  sidereal  day  is  the  interval  between  two  successive 
transits  of  the  vernal  equinox  across  the  same  meridian.      Its 
hours  are   numbered   from   o   to   24.     The   sidereal   time   is 
oh  oom  oos  at  the  instant  when  the  vernal  equinox  transits 
across  the  meridian.     The  sidereal  time  at  a  given  station  and 
instant  is  the  right  ascension  of  the  meridian,  or  is  the  same 
as  the  hour-angle  of  the  vernal  equinox,  counted  in  the  direc- 
tion of  the  apparent  motion  of  the  stars,  at  that  station  and 
instant. 

Right    ascensions    being    reckoned  .from    west    to    east, 


§  ig.  APPARENT  SOLAR    TIME.  1 9 

opposite  to  the  apparent  motion  of  the  stars,  it  follows  from 
the  above  definition  that  the  sidereal  time  at  the  instant  of 
transit  of  a  star  is  the  same  as  the  right  ascension  of  that  star. 

The  sidereal  day  is  substantially  the  interval  of  time 
required  for  one  rotation  of  the  earth  on  its  axis,  and  the 
uniformity  of  the  rotation  of  the  earth  is  depended  upon  to 
furnish  the  ultimate  measure  of  time.  Because  of  the  motion 
of  the  vernal  equinox  on  the  celestial  sphere,  about  50"  per 
year,  the  sidereal  day  and  the  time  of  one  rotation  of  the 
earth  on  its  axis  differ  by  about  one  one-hundredth  of  a 
second. 

19.  The  interval  between  two  successive  transits  of  the 
Sun  across  the  meridian  is  called  an  apparent  solar  day.  The 
apparent  solar  time  for  any  instant  and  station  is  the  hour- 
angle  of  the  Sun,  at  that  instant,  from  that  meridian.  "  But 
the  intervals  between  successive  returns  of  the  Sun  to  the 
same  meridian  are  not  exactly  equal,  owing  to  the  varying 
motion  of  the  Earth  around  the  Sun,  and  to  the  obliquity  of 
the  ecliptic." 

Let  Fig.  i  represent  a  section  of  the  universe  on  the  plane 
of  the  Earth's  orbit  as  seen  from  some  position  in  space  on 
the  side  on  which  the  north  pole  is  situated.  The  Earth  is 
seen  moving  around  its  orbit  in  a  counter-clockwise  direction, 
while  at  the  same  time  its  rotation  about  its  own  axis  appears 
to  be  counter-clockwise.  The  figure  is  not  to  scale,  but  is 
merely  a  diagram  in  which  certain  dimensions  are  exaggerated 
for  the  sake  of  clearness.  Suppose  that  A  is  the  position  of 
the  Earth  at  a  certain  time,  about  March  21,  when  the  Sun 
is  seen  projected  against  the  celestial  sphere  upon  the  vernal 
equinox.  Let  B  be  the  position  of  the  Earth  one  sidereal 
day  later.  Then  Aa  and  Ba  are  parallel  lines,  the  vernal 
equinox  being  at  an  infinite  distance  (on  the  celestial  sphere). 
The  Earth  has  made  one  complete  rotation  on  its  axis  between 


20  GEODETIC  ASTRONOMY.  §  19. 

the  two  positions,  and  the  vernal  equinox  has  returned  to  the 
same  meridian.  The  Earth  having  moved  a  distance  AB 
along  its  orbit,  the  Sun  is  now  seen  projected  against  the 
celestial  sphere  at  b  instead  of  a.  Before  the  Sun  will  return 
to  the  meridian  of  position  A  again  the  Earth  must  rotate 
through  the  additional  angle  represented  by  aBb  reduced  to 
the  plane  of  the  Earth 's  equator.  (The  figure  represents  a 
section  in  the  plane  of  the  ecliptic.}  The  apparent  solar  day 
will  then  be  longer  than  the  sidereal  day  by  the  time  required 
for  the  Earth  to  rotate  through  this  angle, — on  an  average  a 
little  less  than  four  minutes. 

Let  the  angle  governing  the  excess  of  the  apparent  solar 
over  the  sidereal  day  be  examined  further.  As  the  Earth 
proceeds  forward  along  its  orbit  the  Sun  will  apparently  move 
backward  on  the  celestial  sphere  along  the  ecliptic  to  points 
£,  c,  d,  etc.  One  of  the  laws  of  gravitation  governing  the 
motion  of  the  Earth  in  its  orbit  is  that  the  line  joining  the 
Earth  to  the  Sun  sweeps  over  equal  areas  in  equal  times. 
The  linear  velocity  then  varies  nearly  inversely  as  the  distance 
to  the  Sun,  and  the  angular  velocity  varies  still  more  than  the 
linear.  The  angular  velocity  is  about  7$  greater  during  the 
winter  (of  the  northern  hemisphere)  than  during  the  summer. 
The  various  arcs  ab,  be,  cd,  etc.,  along  the  ecliptic,  each 
corresponding  to  one  sidereal  day,  will  vary  in  value  through 
that  range.  But  the  excess  of  the  apparent  solar  over  the 
sidereal  day  depends  upon  these  arcs  projected  upon  the 
equator  along  hour-circles,  the  rotation  of  the  Earth  being 
uniform  when  measured  along  the  equator.  When  such  a 
small  arc  as  ab  near  either  equinox  is  projected  upon  the 
equator,  it  will  be  considerably  reduced,  being  at  an  angle 
of  23^°  to  the  equator, — the  angle  between  the  equator  and 
ecliptic  at  the  equinoxes. 

On  the  other  hand,  when  a  portion  of  the  ecliptic  about 


§  20.  TIME.  21 

midway  between  the  equinoxes  is  projected  along  its  limiting 
hour-circles  upon  the  equator,  the  projected  length  will  be 
greater  than  the  original.  In  short,  the  difference  between 
the  sidereal  and  apparent  solar  day  varies  by  a  rather  compli- 
cated law  from- about  4m  26s  to  3m  35%  being  on  an  average 
3ra  56s.  55  5  (in  sidereal  time). 

20.  Apparent  solar  time  is  a  natural  and  direct  measure  of 
duration,  inasmuch  as  it  is  indicated  directly  by  the  hour- 
angle  of  the  Sun,  the  most  conspicuous  of  all  the  heavenly 
bodies.  But  a  clock  or  chronometer  cannot  be  regulated  to 
keep  this  kind  of  time  accurately,  since  the  different  days  are 
of  unequal  length.  To  avoid  the  difficulties  thus  arising  from 
the  direct  use  of  the  Sun  as  a  measure  of  time,  a  fictitious 
mean  Sun  is  used.  The  mean  Sun  is  supposed  to  move  in  the 
equator  with  a  uniform  angular  velocity,  and  to  keep  as  near 
the  real  Sun  as  is  consistent  with  perfect  uniformity  of  motion. 
This  mean  Sun  makes  one  complete  circuit  around  the 
equator  at  a  uniform  rate  while  the  Earth  is  making  a  com- 
plete circuit  around  its  orbit,  at  a  variable  rate.  It  is  some- 
times as  much  as  16  minutes  ahead  of  the  real  Sun,  and 
sometimes  behind  it  by  that  amount.  A  mean  solar  day  is 
the  interval  between  successive  transits  of  the  mean  Sun  over 
the  same  meridian.  The  mean  solar  time  for  any  instant  and 
station  is  the  hour-angle  of  the  mean  Sun  at  that  instant  from 
that  meridian.  For  brevity  mean  solar  time  is  often  called 
simply  mean  time.  The  mean  solar  day  is  about  3m  56"  longer 
than  the  sidereal  day, — that  being  the  amount  by  which  the 
apparent  solar  day  exceeds  the  sidereal  day  on  an  average. 
Stated  more  exactly,  24  hours  of  mean  solar  time  is  the  same 
interval  as  24h  03™  $6\$$$  of  sidereal  time. 

The  sidereal  and  mean  solar  time  coincide  for  an  instant 
about  March  2 1  each  year.  The  former  gains  24  hours  on 
the  latter  in  a  year. 


22  GEODETIC  ASTRONOMY.  §  21. 

The  equation  of  time  is  the  correction  to  be  applied  to 
apparent  time  to  reduce  it  to  mean  time.  It  is  the  interval 
of  time  by  which  the  mean  Sun  precedes  or  follows,  or  is  fast 
or  slow  of,  the  real  Sun  at  a  given  instant.  Its  limiting  values 
are  about  -f-  i6m  and  —  i6m.  It  is  given  in  the  American 
Ephemeris  and  Nautical  Almanac  for  every  noon  at  Washing- 
ton (and  Greenwich).  It  can  be  obtained  for  any  intermediate 
instant  with  an  error  not  greater  than  os.  I,  usually  much  less, 
by  a  simple  straight-line  interpolation. 

21.  The  civil  day,  according  to  the  customs  of  society, 
commences  and  ends  at  midnight.  The  hours  from  midnight 
to  noon  are  counted  from  o  to  12  and  are  marked  A.M. 
The  remaining  hours  from  noon  to  midnight  are  again  num- 
bered from  o  to  12  and  marked  P.M. 

The  astronomical  day  commences  at  noon  on  the  civil  day 
of  the  same  date.  Its  hours  are  numbered  from  o  to  24, 
from  noon  of  one  day  to  noon  of  the  next.  The  astronomical 
time  as  well  as  the  civil  time  may  be  either  apparent  solar  or 
mean  solar.  The  convenience  of  the  astronomical  day  for  the 
astronomer  arises  from  the  fact  that  he  does  not  have  to 
change  the  date  on  his  record  of  observations  in  the  midst  of 
a  night's  work  as  he  would  be  obliged  to  if  he  used  civil 
dates. 

The  zeros  of  sidereal,  apparent  solar,  and  mean  solar  time 
are,  by  definition,  the  instants  of  transit,  across  the  meridian, 
of  the  vernal  equinox,  the  Sun,  and  the  mean  Sun,  respec- 
tively. The  time,  therefore,  (of  any  of  the  three  kinds,)  will 
be  the  same  for  two  stations  at  a  given  instant  only  in  case 
those  stations  are  on  the  same  meridian.  If  the  stations  are 
not  on  the  same  meridian  the  difference  of  their  times  (of 
any  of  the  three  kinds)  is  a  difference  of  two  hour-angles 
measured  from  the  respective  meridians  to  the  same  object, 
and  is  therefore  the  angle  between  the  meridians  or  the  differ- 


§22.  TIME.  23 

ence  of  longitude  of  the  two  stations.  A  difference  of  longi- 
tude is  then  a  difference  of  time. 

22.  In  the  United  States,  excluding  Alaska,  for  every 
mile  of  distance,  east  or  west  along  a  parallel  of  latitude,  the 
longitude  changes  by  about  four  or  five  seconds  of  time.  If 
each  city  and  town  used  its  own  local  mean  solar  time,  the 
traveller  would  find  himself  at  considerable  inconvenience,  on 
the  modern  railroads  which  transport  him  from  500  to  1000 
miles  per  day,  to  keep  his  watch  regulated  to  the  time  of  his 
various  stopping  points.  Even  when  the  railroads  and  the 
general  public  used  one  particular  time  for  considerable  areas, 
— that  time  being  usually  that  of  some  large  city  or  important 
railroad  division  terminus, — as  was  the  case  a  few  years  ago, 
there  was  still  confusion  and  annoyance  arising  from  the  fact 
that  each  kind  of  time  was  changed  to  the  next  by  the  addi- 
tion or  subtraction  of  some  irregular  number  of  minutes, 
which  was  apt  to  be  forgotten  when  most  needed.  These, 
and  other  reasons,  have  led  to  the  general  adoption  in  this 
country  of  what  is  called  standard  time.  The  standard  time 
for  each  particular  locality  is  the  mean  solar  time  of  the 
nearest  meridian  which  is  an  exact  whole  number  of  hours, 
four,  five,  six,  seven,  etc.,  west  of  Greenwich.  The  standard 
meridians  for  this  country  are  thus: 

75°  or  5h  west  of  Greenwich,  running  near  Utica,  N.  Y., 
Philadelphia,  Pa.,  and  off  Cape  Hatteras. 

90°  or  6h  west  of  Greenwich,  running  near  St.  Louis, 
Memphis,  and  New  Orleans. 

105°  or  7h  west  of  Greenwich,  running  near  Denver, 
Colorado. 

120°  or  8h  west  of  Greenwich,  running  along  the  east  line 
of  the  northern  part  of  California  and  near  Santa  Barbara,  Cal. 

135°  or  9h  west  of  Greenwich,  running  near  Sitka,  Alaska. 

To  reduce  the  local  mean  solar  time  to  standard  time  it 


24  GEODETIC  ASTRONOMY.  §  23. 

is  merely  necessary  in  each  case  to  apply  as  a  correction 
the  difference  of  longitude  of  the  station  and  the  standard 
meridian.  Only  the  astronomer  or  engineer,  however,  is 
obliged  to  use  this  process.  The  traveller  has  occasion 
simply  to  change  from  one  kind  of  standard  time  to  another 
which  differs  from  it  by  exactly  one  hour, — an  interval  which 
is  easy  to  remember. 

To  Convert  Mean  Solar  to  Sidereal  Time. 

23.  To  convert  mean  solar  to  sidereal  time  or  vice 
versa,  it  is  necessary  to  take  account  logically  of  two  facts, 
that  the  zeros  of  the  two  kinds  of  day  differ  by  a  certain 
interval,  to  be  derived  from  the  Ephemeris,  and  that  the  two 
kinds  of  hours  bear  a  fixed  ratio  to  each  other  which  is  nearly, 
but  not  quite,  unity. 

The  local  mean  solar  time  at  St.  Louis,  Mo.,  52™  378.o/ 
west  of  Washington,  is  Qh  2im  23S.35  A.M.,  July  29,  1892. 
What  is  the  local  sidereal  time  ? 

Local  mean  solar  time =          gh  2im  23". 35 

Time  of  mean  noon =        12   oo    oo  .00 


Mean  solar  interval  to  nearest  mean  noon =          2    38     36.65 

Reduction  to  sidereal  interval  (see  §  290) =  -f-    o  oo    26  .06 


Sidereal  interval  to  nearest  mean  noon 2    39    02.71 

Sidereal  time  of  mean  noon,  July 

29,  1892,  at  Washington =  8h  3im  14". 23 

Correction  due  to  longitude  to  re- 
duce to  St.  Louis  (see  below). ...  =  -f-  o  oo  08  .64 

Sidereal  time  of  mean  noon,  July 

29,  1892,  at  St.  Louis =         8    31     22.87 

Required  sidereal  time  at  St.  Louis =          5    52     20  . 16 

The  first  step  is  to  obtain  the  mean  solar  interval  between 
the  given  time  and  the  nearest  mean  noon,  and  to  reduce  it 
to  an  equivalent  sidereal  interval  by  use  of  the  tables  in  §  290 


§  23.  CONVERSION  OF   TIME.  2$ 

(reprinted  from  the  back  part  of  the  Ephemeris).  The 
derivation  of  these  tables  from  the  equation  given  at  the  end 
of  each  is  sufficiently  obvious. 

The  next  step  is  to  derive  from  the  American  Ephemeris 
and  Nautical  Almanac  the  sidereal  time  of  that  mean  noon, 
or,  in  other  words,  the  difference  of  the  zero  points  of  the  two 
kinds  of  time  at  noon  of  that  day.  The  Ephemeris,  in  the 
part  headed  "  Solar  Ephemeris  "  (pp.  377-384  in  the  volume 
for  1892),  gives  directly  the  sidereal  time  of  every  Washington 
mean  noon  for  the  year.  What  is  required  is  the  sidereal 
time  of  St.  Louis  mean  noon.  The  vernal  equinox,  marking 
the  zero  of  sidereal  time,  shifts  3m  56S.555  per  mean  solar  day 
with  respect  to  the  mean  Sun,  marking  the  zero  of  mean  solar 
time.  The  sidereal  time  of  mean  noon  for  a  given  point  then 
increases  3m  563.555  per  day.  St.  Louis  being  52™  378.O7 
west  of  Washington,  its  mean  noon  occurs  at  that  interval  of 
mean  solar  time  later  than  the  mean  noon  of  Washington. 
Its  sidereal  time  of  mean  noon  is  evidently  that  of  Washing- 
ton increased  by  the  motion  of  the  vernal  equinox  relative  to 
the  mean  Sun  in  52™  37S.O7,  or 


This  proportional  part  is  precisely  that  given  by  the  table, 
§  290,  for  the  reduction  of  mean  solar  to  sidereal  time,  and 
hence  the  correction  is  taken  directly  from  that  table. 

Having  now  the  sidereal  interval  to  the  nearest  local  mean 
noon,  and  the  local  sidereal  time  of  that  mean  noon,  the 
required  sidereal  time  is  obtained  by  a  simple  subtraction  (or 
addition,  as  the  case  may  call  for). 

Note  that  the  longitude  of  the  station  is  used  only  in 
reducing  the  sidereal  time  of  mean  noon  at  Washington  to  the 
local  sidereal  time  of  mean  noon.  An  error  of  4s  in  the  longi- 
tude produces  an  error  of  only  O8.oi  in  this  reduction. 


26  GEODETIC  ASTRONOMY.  §  2$. 

Example  of  the  Reduction  from  Sidereal  to  Mean  Time. 

24.  At  a  certain  instant  in  the  evening  of  May  21,  1892, 
at  Harvard  Observatory,  it  was  found  by  an  observation  upon 
a  star  that  the  sidereal  time  was  I3h  4im  27s. 34.  What  was 
the  mean  time  at  that  instant?  Harvard  Observatory  is  23m 
41"  east  of  Washington. 

Given  sidereal  time =        i3h  41™  27". 34 

Sidereal  time  of  mean  noon, 

May  21,  1892,  at  Washington  =  3h  59™  H8.73 

Correction,  due  to  longitude,  to 
reduce  to  Harvard  Obser- 
vatory (§  290) =  —  o  oo  03  .89 


Sidereal    time    of  mean    noon,    May    21,   1892,  at 

Harvard  Observatory '=          3    59    07.84 


Sidereal  interval  after  mean  noon =          9   42     19.50 

Reduction  to  mean  time  interval  (§  291) =  —    o   or     35.40 


Required  mean  time  at  Harvard  Observatory =          9   40    44  .10  P.M. 

The  Ephemeris, 

25.  The  American  Ephemeris  and  Nautical  Almanac 
referred  to  in  the  above  computation  is  an  annual  publication 
of  the  United  States  Government.  It  can  be  obtained  at  any 
time  by  sending  one  dollar  to  the  Nautical  Almanac  Office, 
Washington,  D.  C.  It,  or  its  equivalent,  is  a  necessity  to  an 
engineer  making  astronomical  determinations,  as  will  be  seen 
by  the  many  references  to  it  in  the  following  chapters.  As  it 
forms  a  part  of  the  outfit  of  the  astronomical  observer  and 
computer,  the  student  should  become  familiar  with  its  general 
arrangement,  should  acquire  a  general  understanding  of  all 
parts  of  it,  and  should  obtain  a  thoro'ugh  grasp  of  those  par- 
ticular portions  to  which  he  finds  especial  reference  in  the 
text  of  this  book.  To  gain  familiarity  with  the  most 
frequently  used  portions  of  the  Ephemeris,  it  is  especially 
desirable  that  the  following  pages  of  the  text  at  the  back  of 


§  26.  QUESTIONS  AND    EXAMPLES.  2/ 


the  Ephemeris  headed  "  On  the  Arrangement  and  Use  of  the 
American  Ephemeris"  be  read;  viz.,  the  first  four  pages  of 
the  explanation  of  Part  I  (pp.  493-496  in  the  volume  for 
1892),  and  the  first  three  pages  of  the  explanation  of  Part  II 
(pp.  501-503  of  the  volume  for  1892).  The  Governments 
of  Germany,  France,  and  England,  and  some  others,  issue 
similar  publications. 

QUESTIONS  AND   EXAMPLES. 

26.  i.  The  position  of  the  Sun  projected  upon  the  celes- 
tial sphere  is  always  at  some  point  of  the  ecliptic.  Explain 
why  this  statement  is  not  true  in  regard  to  a  planet. 

2.  What  is  the  relation  between  the  latitude  of  a  station 
and  the  altitude  of  the  pole  at  that  station  ? 

3.  Given  the  latitude  of  a  station  and  the  declination  of 
a  star,  how  may  the  zenith  distance  of  the  star  at  the  instant 
of  upper  culmination  be  determined  ? 

4.  In  the  case  of  a  circumpolar  star  how  may  the  zenith 
distance  at  lower  culmination  be  determined,  the  declination 
and  latitude  being  given  ? 

A  circumpolar  star  is  one  comparatively  near  the  pole,  say 
within  ten  degrees. 

5.  How  would  you  determine  the  zenith  distance  at  upper 
culmination,  and  also  at  lower,  for  a  circumpolar  star  of  which 
the  polar  distance  is  given  ?     The  latitude  of  the  station  is 
supposed  to  be  known. 

6.  The  hour-angle  of  the  star  Vega,  east  of  the  meridian, 
at  a  certain  instant  on  the  evening  of  June  30,  1892,  at  the 
Cornell  Observatory  was  2h  I  im  14s.     The  right  ascension  of 
Vega  at  that  instant  was  i8h  33m   19s.     What  was  the  local 
sidereal  time?    Also,  what  was  the  Washington  sidereal  time, 
—  Cornell  being  2m  i6s  east  of  Washington  ? 


28  GEODETIC  ASTRONOMY.  §  26. 

7.  At  a  certain  instant  the  hour-angle  and  zenith  distance 
of  a  star  are  observed.     The  declination  of  the  star  is  known. 
In  the  spherical  triangle  star-zenith-pole  what  parts  are  known 
and  how  may  the  latitude  of  the  station  be  computed  ? 

8.  What  was  the  hour-angle  of  the  Sun  on  September  29, 
1892,  at  a  station  4h  I9m  46s.  3  west  of  Washington,  when  a 

clock    which    was    3 Is. 9   fast    of   local  mean  time   indicated 
2h  4Im  l8s<9  P.M.? 

The  equation  of  time  for  apparent  noon  at  Washington 
on  September  29  was  —  9™  57s./i  and  for  the  3Oth,  —  iom 
I7S.05.  Ans.  2h  5<Dm  5os.5  west  of  the  meridian. 

9.  The  mean  time  was  5h   i6m  2is.34  P.M.,  August   10, 
1892,  at  a  station  2h  igm  31"  west  of  Washington.     What  was 
the  sidereal  time?     The  sidereal  time  of  mean  noon  at  Wash- 
ington on  that  day  was  gh  i8m  32 '.9 1. 

Ans.    I4h  36m  O9S.  14. 

10.  The  sidereal  time  was  23h  49™  59s. 92,  the  astronomical 
date  August  21,    1892,  and  the  station   ih  29™  2is  west  of 
Washington.     What  was  the  mean  time  and  the  civil  date  ? 
The  sidereal  time  of  mean  noon  at  Washington  on  the  2ist 
(civil  date)  was  iohoim  55S.O2,  and  on  the  22d,  ioh  O5m  5is.57. 
Ans.  Mean  time  =  ih  45m  34s.6o  A.M.     Civil  date,  August  22. 

11.  The  apparent  solar  time  at  a  station  ih  46™  i8s.2  west 
of  Washington  was  at  a  certain  instant  on  April   17,   1892, 
IQh  ^m  i^*f  $  A.M>     What  was  the  mean  time?    The  equation 
of   time  at  Washington    apparent   noon   on    that   date    was 
—  38S.95,  and  on  the  i8th  was  —  52S.49. 

Ans.   ioh  32m  35S.2  A.M. 

12.  The  hour-angle  of  the  Sun  as  observed  at  a  certain 
instant,  at  a  station  2h  14™  34"  east  of  Washington,  on  the 
forenoon  of  May  21,    1892,  was  found  to  be  2h  48*   19".  3. 
What    was   the   sidereal    time  ?     The  equation  of   time  for 
apparent  noon  at  Washington  was  —  3m  3 7s. 96  on  May  20, 


§  26.  QUESTIONS  AND   EXAMPLES.  29 

and  —  3m  33S.96  on  the  2ist.     The  sidereal  time  of  mean 
noon  at  Washington  on  the  2ist  was  3h  59m  ns.73. 

Ans.    IL  o6m  27s. 3. 

13.  Suppose  you  are  carrying  a  watch  which  is  2O8  fast  of 
standard  (75th  meridian)  time,  and  that  you  wish  to  start  a 
sidereal  clock  on  correct  time  to  within  Is  at  Cornell  (2m  i68 
east  of  Washington  or  5h  O5m  56s  west  of  Greenwich).     Sup- 
pose the  date  to  be  Sept.  30,  1892,  and  the  sidereal  time  of 
mean  noon  for  Washington  on  that  date  to  be  I2h  39™  37s. 2. 
What  time  should  the  sidereal  clock  indicate  when  your  watch 
reads  f  oom  oo8?  Ans.    19*  34m  29". 

14.  Explain  why  the  "  sidereal  time  of   mean  noon"  as 
given  in  the  last  column  of  the  "  Solar  Ephemeris "   (pp. 
377-384  in  the  volume  for  1892),  in  the  American  Ephemeris 
and  Nautical  Almanac  is  not  the  same  as  the  "apparent  right 
ascension  "  at  "  mean  noon  "  as  given  in  the  second  column. 

15.  Why    is    not    the    "  equation  of  time    for   apparent 
noon  "  as  given  in  the  eighth  column,  the  same  as  the  differ- 
ence of  the  two  columns  mentioned  in  the  preceding  example? 

1 6.  What  is  the  relation  between  the  right  ascension  of 
the  Sun  at  mean  noon,  the  equation  of  time  at  mean  noon, 
and  the  sidereal  time  of  mean  noon  ? 

17.  Look  up  the  sidereal  time  of  mean  noon  for  to-day 
in  the  Ephemeris.     Then,  knowing  the  time  of  day  and  your 
latitude,  hold  two  sheets  of  paper  parallel  respectively  to  the 
plane  of  the  equator  and  the  plane  of  the  ecliptic. 


30  GEODETIC  ASTRONOMY.  §  2 7. 


CHAPTER   II. 

COMPUTATION    OF   RIGHT   ASCENSION   AND 
DECLINATION. 

27.  In  the  astronomical  practice  of  the  engineer  the  right 
ascension  and  declination  of  the  object  observed  are  usually 
known  quantities  determined  from  sources  external  to  his  own 
observations.  The  object  of  this  chapter  is  to  show  how  the 
right  ascension  and  declination  for  the  instant  of  observation 
are  obtained  from  the  available  sources  of  information. 

The  various  heavenly  bodies  which  the  engineer  is  called 
upon  to  observe  have  all  been  observed  frequently  at  the 
various  fixed  observatories  with  large  instruments  and  at 
many  different  times  extending  over  a  long  period  of  years. 
From  these  observations  the  positions,  that  is,  right  ascen- 
sions and  declinations,  at  various  stated  times  are  determined, 
and  the  motions  are  carefully  computed.  This  makes  it  pos- 
sible to  compute  the  position  of  each  of  these  bodies  at  any 
stated  future  time  with  an  accuracy  depending  on  the  pre- 
cision of  the  observations  and  the  remoteness  of  the  future 
time.  The  results  of  such  computations  of  positions  made 
in  advance,  and  also  the  data  for  such  computations,  are 
given  in  the  ephemerides  issued  by  various  governments:  the 
American  Ephemeris,  Berliner  Jahrbuch,  Connaissance  du 
Temps  (Paris),  British  Nautical  Almanac,  etc.  Various  other 
occasional  publications  also  give  the  data  for  such  computa- 
tions. The  engineer  uses  these  computations  of  position 
made  in  advance,  and  the  published  data  for  such  computa- 


§  2Q.  INTERPOLA  TION.  3 1 

tions,   to  obtain  the  right  ascension  and  declination  at  the 
instant  of  his  observation. 

When  references  are  given  in  the  following  text  to  data  in 
the  American  Ephemeris,  it  should  be  understood  that  sub- 
stantially the  same  data  may  also  be  obtained  from  the  other 
national  ephemerides. 

Position  of  the  Sun  and  Planets. 

28.  The  right  ascension  and  declination  of  the  Sun  are 
given    for    Washington    mean    and    apparent    noon    in    the 
American  Ephemeris  (pp.  3/7-384  of  the  volume  for   1892) 
for  every  day  of  the  year,  together  with  some  other  data  that 
are  frequently  needed  for  computation  purposes.     The  corre- 
sponding data  are  also  given  for  Greenwich  in  first  part  of 
the  Ephemeris.     The  right  ascension  and  declination  are  also 
given  in  the  American  Ephemeris  for  each  planet  for  every 
day  of  the  year  when  its  transit  is  visible  at  Washington  (on 
pp.  393-411  of  volume  for   1892).     The  corresponding  data 
are  given  in  more  complete  form  for  Greenwich  in  the  first 
part  of  the  Ephemeris  (pp.  218-249  °f  tne  volume  for  1892). 
For  the  methods  by  which  the  right  ascension  and  declination 
of  the  Sun,  or  a  planet,  at  any  given  intermediate  time,  are 
to  be  derived  from  the  values  stated  in  the  Ephemeris,  see 
the  following  sections,  Nos.  29-34. 

Interpolation. 

29.  By  interpolation  is  meant  the  process  by  which,  hav- 
ing given  a  series  of  numerical  values  of  a  function  corre- 
sponding each  to  a  stated  value  of  the  independent  variable, 
the  value  of  the  function  for  any  other  intermediate  value  of 
the  variable  is  found  independently  of  a  knowledge  of  the 
analytical  form  of  the  function.     The  independent  variable  is 
often  called  the  argument.     For  example,  the  right  ascension 


32  GEODETIC  ASTRONOMY.  §  30. 

of  the  Sun  is  a  known  function  of  time  as  the  independent 
variable.  It  is  given  in  the  Ephemeris  for  certain  stated 
times.  When  it  is  required  for  any  other  time,  instead  of 
computing  it  directly  from  the  known  function,  it  is  much 
more  convenient  and  rapid  to  deduce  it  by  interpolation  from 
the  stated  numerical  values. 

Interpolation  always  leads  to  approximate  results  which 
may  be  made  more  exact  as  the  process  of  interpolation  is 
made  more  complicated  and  laborious.  The  error  of  inter- 
polation is  the  difference  between  an  interpolated  value  and 
the  value  which  would  be  found  if  one  resorted  to  direct 
computation  from  the  known  function.  Of  the  multitude  of 
methods  of  interpolation,  with  widely  varying  degrees  of  con- 
venience, rapidity,  and  accuracy,  three  methods  will  be  found 
sufficient  for  the  ground  covered  by  this  book.  These  three 
may  be  described  briefly  as  interpolation  along  a  chord,  inter- 
polation along  a  tangent,  and  interpolation  along  a  parabola. 

Interpolation  along  a  Chord. 

30.  In  interpolation  along  a  chord  the  rate  of  change  of  the 
function,  between  the  two  stated  values  of  the  variable  which 
are  adjacent  to  the  value  for  which  the  interpolation  is  to  be 
made,  is  assumed  to  be  constant  and  equal  to  the  total  change 
of  the  function  between  those  points  divided  by  the  interval 
between  the  stated  values  of  the  variable.  If  the  actual 
values  of  the  function  were  represented  graphically,  all  inter- 
polated values  would  lie  along  chords  of  the  function  curve, 
connecting  points  on  the  curve  corresponding  to  stated  values 
of  the  variable.  For  example,  the  right  ascension  of  Jupiter 
at  I2h  35m.5  mean  time  at  Washington,  on  Oct.  i,  1892,  was 


(Ephemeris,   p.   405).      Required  its  right  ascension   at    I5h 
I4m.2   Washington   mean    time,    on   Oct.    i?      The   interval 


§31-  INTERPOLATION.  33 

between  stated  values  of  the  variable  is  23h  55m.6  =  23^93. 
The  change  in  the  value  of  the  function  is  —  298.oi.  The 
rate  of  change  is  then  —  298.oi  -r-  23".  93  =  —  I8.2I2  per 
hour.  The  interval  over  which  the  interpolation  is  carried 
from  the  nearest  given  value  is  I5h  I4m.2  —  I2h  35™.$  =  2h 
38m.7  =  2h.64.  The  change  during  that  interval  is  (2.64) 
(—1.212)  =  —3s.  20.  The  required  right  ascension  is  ib  2im 
078.8i  —  3'.  20  =  ih  2im04s.6i.  The  result  would  have  been 
identical  with  this  had  the  interpolation  been  made  from  the 
other  adjacent  value,  namely,  that  at  I2h  31™.  ion  Oct.  2d. 
In  algebraic  form  this  interpolation  maybe  expressed  thus: 

Vr—V 


.      or          Fl=Ft-(Ft-Ffj.     .     .      .     (i) 

the  first  form  being  used  when  the  interpolation  is  made 
forward  from  the  value  Flt  and  the  second  when  it  is  made 
backward  from  F.2. 

Fj  is  the  required  interpolated  value  corresponding  to  the 
value  F/  of  the  independent  variable.  Vl  and  F,  are  the 
adjacent  stated  values  of  the  argument  to  which  correspond 
the  given  values  Fl  and  F9  of  the  function. 

Interpolation  along  a  Tangent. 

31.  Interpolation  along  a  tangent  is,  in  general,  more 
accurate  than  interpolation  along  a  chord,  but  can  only  be 
used  conveniently  when  the  rates  of  change,  or  first  differen- 
tial coefficients  of  the  function,  are  given  at  the  stated  values 
of  the  variable,  in  addition  to  the  values  of  the  function 
itself.  In  this  interpolation  the  rate  of  change,  for  the  inter- 
val from  the  nearest  stated  value  of  the  variable  to  the  value 


34  GEODETIC  ASTRONOMY.  §  31. 

for  which  the  interpolation  is  to  be  made,  is  assumed  to  be 
constant  and  equal  to  the  given  rate  of  change  at  the  stated 
value  of  the  variable. 

The  interpolated  points  represented  graphically  would  lie 
on  a  tangent,  at  the  nearest  stated  value  of  the  variable,  to 
the  curve  representing  the  function.  For  example,  let  it  be 
required  to  find  the  declination  of  the  Sun  on  Sept.  5,  1892, 
at  9h  30m  A.M.,  Washington  mean  time.  On  page  382  of  the 
Ephemeris  for  that  year,  the  nearest  time  for  which  the 
declination  is  given,  is  Washington  mean  noon  of  that  day. 
For  that  instant  the  declination  is  +  6°  28'  i8".6.  Its  rate 
of  change  for  that  instant  is  stated  to  be  —  5  5".  92  per  hour. 
The  interval  over  which  the  interpolation  is  to  extend  is  2h.5 
backward  from  noon.  Then  by  interpolation  along  the  tan- 
gent to  the  curve  (representing  declinations)  at  noon  of  Sept. 
5th,  there  is  obtained  as  the  declination  at  9h  30™  A.M., 
6°  28'  i8".6  +  (2.5)(55//.92)  =  6°  30'  38".4.  In  this  method 
of  interpolation  the  shorter  the  tangent  the  smaller  the  error 
of  interpolation,  and  therefore  care  should  be  taken  to  inter- 
polate from  the  nearest  stated  value  of  the  variable.  The 
formula  for  this  interpolation  is 


Fj  and  Vj  are  the  required  interpolated  value  and  the  corre- 
sponding given  argument,  Vl  and  F:  are  the  nearest  tabular 
value  of  the  argument  and  the  corresponding  value  of  the 

function,  and  (-77?)  is  the  given  first  differential  coefficient 
corresponding  to  V^ 


§33-  INTERPOLATION.  35 

Interpolation  along  a  Parabola. 

32.  In  interpolation  along  a  parabola  it  is  assumed  that  the 
second  differential    coefficient    of    the    function    is    constant 
between  adjacent  stated  values  of  the  independent  variable, 
or,  in  other  words,  that  the  rate  of  change  of  slope  of  the 
function    curve    is    constant    between    those    points.       This 
assumption  places  the  interpolated  points  along  a  parabola, 
with  axis  vertical,  passing  through  two  points  of  the  function 
curve, — the  uniform  rate  of  change  of  slope  being  a  property 
of  such  a  parabola.     There  are  two  cases  arising  under  this 
method,  depending  upon  whether  the  first  differential  is,  or 
is  not,  given  for  the  stated  values  of  the  variable. 

33.  For  an  example  of  the  first  case  take  the  problem 
proposed  in  the  preceding  section,  in  which  it  is  required  to 
find  the  declination  of  the  Sun  at  gh  30™  A.M.,  Washington 
mean  time,  Sept.  5,  1892.     The  data  given  in  the  Ephemeris 
for   1892   for  Washington  mean  noon  Sept.  4  are  declination 
=  +6°  50'  37". 5,  and  the  first  differential  coefficient  =  — 
55".66;  and  for  Sept.  5,  declination  =  6°  28'  i8".6,  and  first 
differential  coefficient  =  —  5 5". 92.      It  is  proposed  to  place 
the  interpolated  value  on  a  parabola  (with  axis  vertical)  coin- 
ciding with  the  curve  of  declinations  at  the  two  given  points, 
and  also  having  a  common  tangent  at  each  of  these  points. 
To  make  the  interpolation,  the  principle  will  be   used  that  a 
chord  of  such  a  parabola  is  parallel  to  the  tangent  at  a  point 
of  which  the  abscissa  is  the  mean  of  the  abscissae  of  the  two 
ends  of  the  chord.     The  slope  of  the  chord  (of  the  parabola) 
corresponding  to  the  interval  9h  3Om  to   I2h,  on   Sept.    5,  is 
then  the  same  as  the  slope  of  the  tangent  at  the  middle  of 
that  interval,    ioh  45™.     The   slope  of  the   tangent  changes 
by  (—  5 5". 92)  —   (—  55".66)  =  —  o".26  in  24  hours,    or 
—  o".oio8   per  hour.     The    slope    at    ioh  45°*  =  —  $$".92 


36  GEODETIC  ASTRONOMY.  §  34. 

+  (c/.oio8)(i.25)  =  —  55".9i.       The   interpolated  value  at 
9h  3om  is  6°  28'  i8".6  +  (55".9i)(2.5)  =  6°  30''  38".4. 

This  method  of  interpolation,  though  most  easily  remem- 
bered, perhaps,  in  the  geometrical  form,  may  be  put  in  con- 
venient algebraic  form  as  follows: 


dF\  ((dF\         ldF\   1    fi(F>-  F,)l  ~1 


or 

Fi  = 


according  to  whether  the  interpolation  is  made  forward  from 
Ft  or  backward  from  F2.  The  notation  is  the  same  as  in  the 
preceding  paragraphs.  The  two  results  are  identical,  but  the 
arithmetical  work  will  be  shorter  if  the  interpolation  is  made 
from  whichever  of  the  given  points  happens  to  be  the  nearer. 
34.  The  second  case  of  interpolation  along  a  parabola 
occurs  when  the  first  differential  coefficients  are  not  given. 
The  assumptions  involved  are  just  as  before.  As  an  example, 
take  the  problem  proposed  a  few  paragraphs  back,  of  finding 
the  right  ascension  of  Jupiter,  at  I5h  14™. 2,  Washington  mean 
time,  on  Oct.  i,  1892.  The  Ephemeris  gives  the  right 
ascension 

=  ih  2im  368.59  at  I2h  39m.9  on  Sept.  30; 

=  ih  2im  o;8.8i  at  I2h  35m.5  on  Oct.  i ; 

=  ih  20m  38s.8o  at  I2h  31™.  i  on  Oct.  2. 

It  is  proposed  to  interpolate  the  required  point  on  a  pa- 
rabola, with  axis  vertical,  passing  through  these  three  given 

r  (dF\       fdF 
*  If  the  second  derivative  is  constant,  then 


ridF\ 

- 

\_\dVh 


is  really ,.     Call  Vi  —  V\*  dV.     Then  (3)  put  in  the  calculus  notation 


dF 

becomes  FI  =  Fi-\-  — -.  dV ' -\-  ——,  —  '  in  which  — -  and   TTTT  are  values 
dV  dfy*     2  dV  * 

corresponding  to  the  point  F\  .  V\. 


§34-  INTERPOLATION.  37 

points.  Again,  using  the  principle  that  in  such  a  parabola,  a 
chord,  and  the  tangent  at  a  point  of  which  the  abscissa  is  the 
mean  between  the  abscissae  of  the  two  ends  of  the  chord,  are 
parallel,  the  slope  of  the  parabola  at  any  point  may  be  com- 
puted. The  slope  of  the  parabola  at  the  middle  of  the  first 
interval,  at  oh  37m-7  =  oh.63  on  Oct.  i,  is  (07". 8 1  —  36". 59) 
-i-  23.93  =  IB.2O3  per  hour.  At  the  middle  of  the  second 
interval,  at  oh  33^3  =  oh.56  on  Oct.  2,  it  is  (388.8o  —  678.8i) 
-f-  23.93  =  —i8. 212  per  hour.  The  interval  over  which  the 
interpolation  is  made,  from  the  nearest  given  value,  is  I2h 
35m.5  to  I5h  I4m.2  on  Oct.  i,  or  2h  38m.7  =  2h.64.  The 
slope  of  the  chord  for  this  interval  is  that  of  the  tangent 
at  its  middle,  I3h  54™. 8  =  I3b.9i.  This  slope  is,  assum- 
ing the  rate  of  change  of  the  slope  constant,  —  I8.2O3  -j- 

s£  56l£l|E<-  I-.2I2)  -  (-  I-.203)]  =  -  I-.208  per 
hour.  The  right  ascension  at  I5h  I4m.2  is  ih2imO78.8i  — 
( i8. 208X2.64)  =  ih  2im  04s. 62.  This  sample  interpolation  is 
made  in  the  present  form  simply  for  the  purpose  of  illustrat- 
ing the  principles  involved.  The  numerical  work  of  inter- 
polation should  ordinarily  be  done  as  indicated  in  formula  (4) 
of  the  following  section. 

Putting  this  method  in  the  algebraic  language  it  takes  the 
following  form:  Let  F^  Fw  and  F%  be  three  successive  given 
values  of  the  function  corresponding  to  the  values  Flf  F3,  and 
Fs  of  the  independent  variable ;  and  let  Fa  be  the  stated  value 
of  the  variable  nearest  to  which  lies  the  value  for  which  the 
interpolation  is  to  be  made.  Let  F2  be  the  required  value  of 
the  function  corresponding  to  Vf.  Then 


•      •  y .    I     rr         y     i    y        • 

/    -Pi+l    Fa_ ^+ 1  r>_ ^3- Fa_ ri  j  Fa_|_ ra     FT+F!    KFi-rO; 

~T~       ~~^~ 


38  GEODETIC  ASTRONOMY.  §  35 

or,  in  simplified  form, 

F—F       ^F—F     F—  F  1  f  Vr—  Fi 

3  l     I     I       s        *^1        -*!|       *l-  l   I  */        *i 


If,  as  is  usually  the  case,  the  successive  differences 
between  F,,  Fa,  and  F8  are  all  the  same  and  equal  to  D,  this 
may  be  further  simplified  to  the  form 


,-  V,\;      (4a) 


in  which  d^  is  the  second  difference,  or  (F^  —  F^  —  (F^  —  F^. 
The  second  term  in  the  square  bracket  will  usually  be  com- 
paratively small,  and  therefore  easy  to  compute. 

For  a  more  complete  discussion  of  interpolation,  giving 
other  more  complex  and  accurate  formulae,  see  Chauvenet's 
Spherical  and  Practical  Astronomy,  vol.  I.  pp.  79-91  ; 
Doolittle's  Practical  Astronomy,  pp.  69-98;  and  Loomis* 
Practical  Astronomy,  pp.  202-212. 

Accuracy  of  Interpolation  of  Position  of  Sun  and  Planets. 

35.  An  interpolation  along  a  tangent,  —  the  first  differen- 
tial coefficients  or  hourly  changes  being  given,  —  from  the 
values  given  for  noon  of  each  day  in  the  Ephemeris  (pp. 
377-384  of  the  volume  for  1892),  will  give  the  right  ascension  of 
the  Sun  at  any  time  with  an  error  of  interpolation  not  exceed- 
ing os.6,  and  the  declination  with  an  error  of  interpolation  not 
exceeding  i/r.8.  For  nearly  all  cases  the  error  of  interpola- 
tion will  be  much  less  than  these  extreme  limits.  Approxi- 
mately, the  extreme  error  of  interpolation  along  a  tangent  is 
one-eighth  of  the  second  difference  at  that  point,*  —  meaning 

*  The    interpolation  along  a  tangent  will  evidently  give  the  greatest 
error  when  the  interpolated  point  is  midway  between  the  tabulated  values, 


§37-  STAR   PLACES.  39 

by  a  second  difference  the  difference  between  successive  first 
differences.  If  greater  accuracy  is  desirable, — which  will 
often  be  true  of  declinations,  but  seldom  of  the  right  ascen- 
sions,— an  interpolation  along  a  parabola  will  always  give  all 
needful  accuracy.  In  dealing  with  the  planets  an  interpola- 
tion along  a  tangent,  or  along  a  chord  in  those  cases  in  which 
the  first  differential  coefficients  are  not  given,  will  in  many 
cases  give  a  sufficient  degree  of  accuracy,  and  interpolation 
along  a  parabola  will  give  all  needful  precision  in  every  case. 

Position  of  the  Moon. 

36.  In    the   first    part   of    the   Ephemeris,   in   which   the 
standard  meridian   is  that  of  Greenwich  (pp.   2-217  °f  the 
volume  for  1892),  the  Moon's  right  acension  and  declination 
are  given  for  every  hour  during  the  year,  together  with  the 
corresponding  first  differential  coefficients.     An  interpolation 
along  a  tangent,  from  the  nearest  hour,  will  give  the  Moon's 
right  ascension  at  any  time  with  an  error  of  interpolation  not 
exceeding  O3.O5.      The   corresponding    limit   for    declination 
interpolated  along  a  tangent  is   i" .     This  will  usually  be  a 
sufficient  degree  of  accuracy.      But  if  for  some  special  reason 
a  greater  precision  is  required,  an  interpolation  along  a  parab- 
ola will  give  the  results  far  within  the  limits  of  error  of  the 
tabular  values  themselves. 

Positions  of  Stars. 

37.  The  American   Ephemeris  gives  the  right  ascension 
and  declination  of  four  close  circumpolar  stars  for  every  upper 

the  tangent  then  used,  corresponding  to  one-half  of  a  tabular  interval, 
being  longer  than  is  necessary  in  any  other  case.  If  for  this  case  the 
interpolation  along  a  parabola  be  used,  the  interpolated  value  will  differ 
from  that  found  by  using  the  tangent  by  one-eighth  the  second  difference, 
— as  may  be  seen  by  inspection  of  the  formulae  (2),  §31,  and  (3),  §  33.  If, 
then,  the  second  interpolation  be  assumed  to  be  exact,  this  value  is  the 
error  of  the  first  interpolation. 


40  GEODETIC  ASTRONOMY.  §  3/. 

transit  at  Washington  (pp.  302-313  of  1892);  of  every  tenth 
transit  for  about  200  stars;  and  the  right  ascension  only  for 
every  tenth  transit  visible  at  Washington  of  about  200  more. 
Other  national  Ephemerides  contain  similar  lists,  which  often 
comprise  about  the  same  stars.  This  list  is  made  up  of  stars 
whose  positions  are  well  determined  by  many  observations  at 
various  observatories.  They  are  also  chosen  with  especial 
reference  to  the  needs  of  the  engineer  and  navigator  as 
regards  brightness  and  distribution  on  the  celestial  sphere. 
An  idea  of  the  care  with  which  their  positions  have  been 
determined  may  be  gained  from  the  mere  statement  of  the 
fact  that  in  computing  many  of  these  declinations  fifty  cata- 
logues of  recorded  observations,  at  many  different  observa- 
tories, made  at  various  times  during  a  total  interval  of  a 
century  and  a  quarter,  were  consulted,  and  the  various 
observations  upon  any  one  star  combined  in  each  case  in  a 
single  least-square  computation.* 

The  positions  of  the  close  circumpolars  at  any  time  may 
be  obtained  with  all  needful  accuracy  by  interpolation  along 
a  chord  from  the  values  given  in  the  Ephemeris.  For  the 
other  stars  given  in  the  Ephemeris  (at  lO-day  intervals)  an 
interpolation  along  a  parabola  will  usually  be  necessary. 

When  other  stars  must  be  observed  than  these  Ephemeris 
stars  of  which  the  places  are  given  at  frequent  intervals,  a 
complicated  procedure  is  necessary  to  obtain  the  position  of 
the  star  at  the  time  of  the  observation.  This  process  forms 
the  subject  of  the  remainder  of  this  chapter. 

The  position  or  place  of  a  star  is  usually  given  in  one  of 

*  See  "Survey  of  the  Northern  Boundary  from  the  Lake  of  the  Woods 
to  the  Rocky  Mountains  "  (Washington,  1878),  pp.  409-615,  for  a  complete 
report  on  the  computation  of  star  places  for  that  survey  by  Lewis  Boss  (pp. 
421-424  give  catalogues  consulted).  Many  of  the  star  places  given  in  the 
Ephemeris  are  from  this  computation. 


§3$.  A  BERRA  TION.  4 1 

three  ways,  which  should  be  carefully  distinguished.  Either 
its  apparent  place,  true  place,  or  mean  place  is  given.  The 
right  ascension  and  declination,  as  defined  in  §  12,  indicate 
the  true  place  of  a  star  or  other  celestial  object.  But  the 
apparent  direction  of  a  star,  even  aside  from  the  refraction  of 
the  line  of  sight  by  the  terrestrial  atmosphere,  is  affected  by 
aberration.  The  apparent  place  of  a  star  is  its  true  place 
modified  by  the  aberration  of  light.  An  observer  sees  a  star 
in  a  position  which  differs  from  what  is  technically  called  its 
apparent  place  by  the  effect  of  refraction  only.*  It  should  be 
carefully  noted  that  the  word  "  apparent  "  is  not  here  used 
in  the  ordinary  sense,  but  in  the  special  technical  sense  which 
it  must  be  understood  to  have  hereafter  throughout  this  book. 

Aberration. 

38.  Aberration  is  an  apparent  displacement  of  a  star 
resulting  from  the  fact  that  the  velocity  of  light  is  not  infinite 
as  compared  with  the  velocity  of  motion  through  space  of  the 
observer,  stationed  at  a  point  on  the  Earth's  surface. 

If  one  is  standing  in  a  rain  which  is  falling  in  vertical 
lines,  the  umbrella  must  be  held  directly  overhead.  If,  how- 
ever, one  is  riding  rapidly  through  such  a  rain-storm,  the 
umbrella  must  be  inclined  forward.  In  the  first  case  a  drop 
of  rain  entering  at  the  centre  of  one  end  of  a  straight  open 
tube  held  with  its  axis  vertical  would  pass  along  the  axis  of 
the  tube  to  the  other  end  without  touching  the  tube.  In  the 
second  case,  however,  if  it  is  desired  that  drops  which  enter 
the  tube  at  the  upper  end  shall  continue  down  the  tube 
without  touching  the  sides,  it  will  be  necessary  to  incline  the 
tube  forward  from  the  vertical  to  a  certain  angle  which  is 
dependent  on  the  relative  velocity  of  the  horizontal  motion 
of  the  tube  and  the  vertical  motion  of  the  rain.  So  when  a 

*  For  a  detailed  consideration  of  refraction  see  §§67-69. 


42  GEODETIC  ASTRONOMY.  §  3Q. 

telescope  is  to  receive  along  its  axis  the  light  undulations  from 
a  star,  it  must  be  inclined  forward  in  the  direction  of  the 
actual  motion  of  the  telescope  in  space  so  as  to  make  a  slight 
angle — the  aberration — with  the  actual  line  joining  telescope 
to  star.  This  small  angle,  the  aberration,  amounting  at 
most  to  about  20",  is  evidently  dependent  upon  the  relative 
velocity  of  light  and  of  the  telescope,  and  the  angle  between 
those  two  velocities.  (The  student  may  easily  draw  a 
diagram  for  himself  showing  the  geometrical  relations  con- 
cerned.) The  motion  of  the  telescope  is  compounded  of  that 
due  to  the  diurnal  rotation  of  the  Earth  on  its  axis  and  the 
annual  revolution  of  the  Earth  about  the  Sun.  These  give 
rise  to  the  diurnal  aberration  and  annual  aberration,  respec- 
tively. The  diurnal  aberration  evidently  affects  right  ascen- 
sions directly,  but  has  no  effect  upon  declinations.  The 
annual  aberration  in  general  affects  both.  For  an  example 
of  the  way  in  which  diurnal  aberration  is  taken  into  account 
in  computations,  see  §  96.  The  effect  of  annual  aberration 
is  included  in  the  apparent  place  computation  created  later  in 
this  chapter. 

The  velocity  of  light  is,  according  to  the  best  determi- 
nations, about  186300  miles  per  mean  solar  second.*  It 
requires  about  eight  minutes  for  light  to  travel  from  the  Sun 
to  the  Earth.  An  observer,  then,  does  not  see  a  celestial 
object  in  its  true  position  at  the  instant  when  the  light  enters 
the  eye,  but  in  the  position  which  it  occupied  when  that  light 
left  the  object — an  appreciable  interval  earlier,  for  all  celestial 
objects.  This  phenomenon  is  called  planetary  aberration. 
With  this  form  of  aberration  the  engineer  is  not  concerned. 

39.  The  mean  place  of  a  star  is  its  position  referred  to  the 
mean  equator  and  mean  ecliptic,  as  distinguished  from  its 

*  See  "  The  Solar  Parallax  and  its  Related  Constants,"  Wm.  Harkness, 
Washington,  1891,  pp.  142  and  29-32. 


§39-  MEAN-  PLACES.  43 

position  as  referred  to  the  actual  or  true  equator  and  ecliptic. 
The  equator  and  ecliptic  as  they  would  be  if  unaffected  by 
periodic  variations,  in  other  words  by  nutation,  are  called  the 
mean  equator  and  mean  ecliptic. 

The  mean  place  of  a  star,  then,  at  a  given  instant,  differs 
from  the  true  place  by  the  effect  of  nutation  at  that  instant, 
and  from  the  apparent  place  by  the  effects  of  both  nutation 
and  aberration. 

To  avoid  inconveniences  arising  in  the  course  of  computa- 
tions of  star  places,  if  any  other  form  of  year  is  employed  in 
reckoning  time,  the  astronomer  uses  what  is  called  the 
Besselian  fictitious  year.  The  beginning  of  the  fictitious  year 
is  the  instant  at  which  the  celestial  longitude  of  the  mean 
Sun  is  280°,  or,  in  other  words,  when  the  mean  Sun  is  280° 
from  the  vernal  equinox  measured  along  the  ecliptic.*  The 
beginning  of  the  fictitious  year  differs  from  the  beginning  of 
the  ordinary  year  by  a  fraction  of  a  day,  which  varies  for 
different  years. 

The  places  given  in  the  Ephemeris,  referred  to  in  §  37, 
for  every  day  or  every  ten  days,  are  apparent  places,  and  are 
so  marked.  When  the  engineer  is  obliged  to  have  recourse 
to  stars  which  are  not  so  given  in  the  Ephemeris,  he  consults 
one  or  more  of  the  various  available  star  catalogues  or  star 
lists. f  These  catalogues  and  lists  give  the  mean  places  of  the 
stars  at  the  beginning  of  some  stated  fictitious  year,  together 
with  other  data  relative  to  each  star.  The  problem  which 
then  confronts  the  engineer  is  to  derive,  from  that  given 
mean  place,  the  apparent  place  at  the  time  at  which  his 
observation  was  made.  This  is  done  in  two  steps.  Firstly, 
the  mean  place  of  the  star  is  reduced  from  the  epoch  of  the 
catalogue  to  the  beginning  of  the  fictitious  year  at  some 

*  See  definition  of  celestial  longitude,  §15. 

f  For  references  to  a  few  of  such  catalogues  and  lists  sec  §  141. 


44  GEODETIC  ASTRONOMY.  §41. 

part  of  which  its  apparent  place  is  desired.  Secondly,  the 
apparent  place  of  the  star  at  the  time  of  observation  is 
deduced  from  the  mean  place  at  the  beginning  of  the  ficti- 
tious year. 

Reduction  of  Mean  Places  from  Year  to  Year. 

40.  To  serve  as  a  concrete  example,  let  it  be  supposed 
that  the  star  ^  Hercules  was  observed  at  its  transit  across  the 
meridian   at   St.    Louis,    Mo.,    on   July    16,    1892;   and    the 
authority  depended  upon  for  its  position  is  Boss's  Catalogue 
of  500  Stars  for  1875.0.*     This  star  is  No.  312  in  that  cata- 
logue and  its  mean  place  as  there  given  for  the  beginning  of 
the  fictitious  year  1875  *s 

^1875.0  =  i?h  4im  34s- o  =  mean  right  ascension; 
#1875.0  —  +  27°  47'  42"'1?  —  mean  declination. 

(Throughout  this  book  a  and  6  will  be  used  to  indicate 
the  apparent  right  ascension  and  declination,  respectively,  at 
the  time  of  the  observation  under  consideration.  The  same 
letters  with  the  subscript  m,  thus,  am,  dm,  will  be  used  to  indi- 
cate the  mean  place.  With  a  year  as  a  subscript  as  above, 
they  will  be  understood  to  indicate  the  mean  place  at  the 
beginning  of  that  fictitious  year.) 

41.  The  reduction  from  the  mean  place  at  1875.0  to  that 
at    1892.0  involves  simply  the  change  in  the  mean  equator 
and  mean  ecliptic  during  that  time.     The  determination  of 
the  laws  of  change  of  these  two  fundamental  reference  circles, 
and  the  method  of  computing  the  effect  of  those  changes  upon 
right  ascensions  and  declinations,  belong  rather  to  the  prov- 
ince of   the  astronomer  than  to  that  of  the  engineer.      It 

*  Survey  of  the  Northern  Boundary  from  the  Lake  of  the  Woods  to  the 
Rocky  Mountains  (Washington,  1878),  pp.  592-615. 


§41.                                            ME  A  N  PL  A  CES.  45 

suffices  for  the  engineer  to   accept  the  results  of  the  investi- 
gations of  the  astronomer  in  the  following  form : 

dam  f  . 

—  =  *»+«.  sin  <*„  tan  *„  +  /!;    ...  (5) 

—£-  =  n  .  cos  am  —  // ; (6) 

d*dm       dn  da 


in  which  m  =  46". 062  3  -f-  o". 0002  849^  —  1800)  (/  being  ex- 
pressed in  years),  and  n  =  20". 0607  —  o//.oooo863(/  —  1800). 
The  numerical  values  for  m  and  n  as  here  given  are  those 
most  extensively  used,  and  are  the  result  of  exhaustive 
investigations  by  the  astronomers  Peters  and  Struve.  /*  and 
//  are  proper  motions  per  year  in  right  ascension  and  declina- 
tion respectively,  for  an  account  of  which  see  §§44,  45.  For 
the  present  these  proper  motions  may  be  considered  simply 
as  changes  at  a  uniform  rate  in  each  of  the  two  co-ordinates, 
without  any  reference  to  their  meaning  or  method  of  deriva- 

dam  ddm 

tion.      —j~  and  -j-  are  rates  of  change  per  year. 

The  formulae  given  above  are  neither  complete  nor  exact, 
many  terms  of  the  exact  formulae  having  been  dropped,  and 
those  which  are  retained  having  been  somewhat  modified. 
But  they  furnish  the  complete  basis  for  a  reduction,  with 
sufficient  accuracy  for  the  purposes  of  the  engineer,  from  the 
mean  place  given  in  a  catalogue  to  the  mean  place  at  the 
beginning  of  any  other  fictitious  year  within  thirty  or  perhaps 
fifty  years.  For  the  formulae  in  complete  form  adapted  to 
the  use  of  the  astronomer  to  bridge  over  long  intervals  of 
time, — sometimes  more  than  a  century, — see  "  Survey  of 
the  Northern  Boundary  from  the  Lake  of  the  Woods  to  the 


46  GEODETIC  ASTRONOMY.  §  42. 

Rocky  Mountains,"  pp.  416-420;  and  for  a  detailed  discus- 
sion of  them  see  Doolittle's  Practical  Astronomy,  pp.  560- 
578  and  583-5^9. 

42.  The  engineer  is,  however,  relieved  of  the  necessity  for 
performing  the  numerical  operations  indicated  by  formulae 
(5),  (6),  and  (7).  For  star  catalogues  and  lists  give  in  addi- 

j  7j\  7  2  j\ 

tion  to  am  and  Sm  the  values  of  —£,  -jr,  and  the  term  -j~ 
is  tabulated,*  in  §  292  of  this  book,  for  the  arguments 
am  and  —ji-,  of  which  it  is  evidently  a  function. 

Students  will  find  slight  differences  between  different 
authorities  in  regard  to  the  nomenclature  of  this  part  of  the 

*  This  table  is,  so  far  as  the  author  knows,  a  new  one.  It  was  com- 
puted from  formula  (7)  above,  for  the  date  1900,  to  six  places  of  decimals 
and  afterward  reduced  to  five.  It  is  hoped  that  all  the  tabular  values  are, 
in  so  far  as  the  computation  is  concerned,  within  0.6  of  a  unit  in  the  fifth 

d*dm 
place.     The  formula  used  for  —7-5-  is,  however,  approximate  in  itself  in 

having  omitted  the  effect  of  proper  motion.  Theory  indicates  that  this 
omission  should  produce  an  error  so  small  as  to  be  negligible  for  our 
present  purpose.  To  test  that  conclusion,  as  well  as  the  accuracy  of  com- 

»2  j> 

putation   of   the  table,          ™   for  fifty   stars  (every  tenth)  of    Boss'    List, 

Northern  Boundary  Report,  was  derived  from  the  table  and  compared 
with  that  given  by  Boss  in  the  list.  Boss'  values  were  computed  from  the 
exact  formulae.  The  greatest  difference  found  was  o". 00003.  This  would 
cause  an  error  of  only  o".oi  in  a  reduction  extending  over  30  years,  and 
only  o".O4  in  50  years.  It  is  believed,  therefore,  that  the  table  is  abundantly 
accurate  within  the  limits  over  which  its  arguments  extend.  It  should 
not,  however,  be  assumed  to  hold  good  beyond  those  limits.  The  table 

does  not  cover  the  comparatively  rare  cases  in  which  —  is  negative  (for 

stars  near  the  pole).  For  these  cases  the  formula  (7)  must  be  used.  The 
table  is  computed  for  the  year  1900.  The  same  computation  made  for  any 
date  between  1700  and  2100  would  give  values  differing  from  those  of  the 
table  by  not  more  than  one  unit  in  the  last  decimal  place  given  in  the 
table. 


§43-  MEAN  PLACES.  47 

•  S 

subject.      — p   is  usually  known  as  the  "  annual  variation  in 

right  ascension/'     —^  is  sometimes  given  under  the  heading 

"  annual    variation   in   declination."     Sometimes   the   terms 
n  cos  a  and  //  are  given  separately  as  "  annual  precession  " 

TJ    » 

and  "  proper  motion,"  respectively.        .  am  is  sometimes  given 

in  the  form  of  a  "  change  per  100  years  "  in  the  annual  pre- 
cession. 

Having  given  the  mean  place  of  a  star,  am  and  6m  for  a 
date  tm  at  the  beginning  of  some  fictitious  year,  the  place  at 
time  t0  at  the  beginning  of  any  other  fictitious  year  is 


(8) 


~(t,  -  of  d^r-  •  (9) 

43.   To  return  to  the  numerical  case  in  hand,  the  following 
data  are  also  given  in  Boss*  list  for  the  star  /*  Hercules: 

~—7j  —  annual  variation  in  right  ascension  =  +  2s-  345  \ 

-r    =  annual  variation  in  declination,  including  proper  motion 


all  for  the  date  1875. 
The  mean  place  for  1892  is  then,  by  (8)  and  (9), 


l892  =  27°  47'  42".  17  +  (i7)(-2//.370i)+Ki7)a(o//.oo338o) 
=  27°  47r  42".  17  -  40".29  +  o".49  =  27°  47'  O2//.37. 


48  GEODETIC  ASTRONOMY.  §  44. 

• 

d*dm 
If  — TT  had  not  been  given  in  the  star  list,  as  frequently  it 

is  not,  it  could  have  been  obtained  by  entering  the  table  §  292 
with  the  arguments  am  =  i?h  4im.6  and  —~  =  -|-  2s. 345. 

The  value  as  then  found  from  the  table  would  have  been 
=  o". 0034 1,  and  the  final  value  for  dl892  would  have  been 
identical  with  that  given  above. 

Proper  Motion. 

44.  When  the  co-ordinates  of  a  star,  as  observed  directly 
at  widely  separated  times,  are  reduced  to  the  same  epoch,  it 
is  usually  found  that,  aside  from  discrepancies  arising  from 
accidental  errors  of  observation,  there  are  systematic  differ- 
ences in  the  various  values  indicating  a  steady  movement  of 
the  star  in  some  one  direction  with  the  lapse  of  time. 
Observations  on  another  star  indicate  usually  that  it  has  also 
such  a  motion  peculiar  to  itself,  which  is  without  any  apparent 
relation  to  the  motion  of  the  first  star.  So  each  star  is,  in 
general,  found  to  have  an  unexplained  motion  peculiar  to 
itself,  called  its  proper  motion.  This  proper  motion  is  always 
exceedingly  small,  and  is  assumed  to  take  place  along  an  arc 
of  a  great  circle  of  the  celestial  sphere,  and  at  a  uniform  rate 
in  each  case.  Probably  neither  of  these  assumptions  are 
strictly  true;  but  the  accumulated  proper  motion  for  several 
centuries  even  would  be  so  small  that  observations  of  the 
highest  degree  of  accuracy  now  obtainable  would  not  be 
sufficient  to  prove  the  path  of  star  to  be  curved,  or  its  motion 
to  be  other  than  uniform. 

When  the  mean  position  of  a  star  for  a  given  date  is  to  be 
derived  from  the  results  of  many  observations  at  various 
times  in  the  past  by  the  process  indicated  briefly  in  §  37,  an 
unknown  annual  proper  motion  in  declination,  and  another  in 


§  45-  PROPER   MOTION.  49 

right  ascension,  are  introduced  into  the  least  square  adjust- 
ment. The  annual  proper  motions  in  declination  and  in 
right  ascension  as  thus  derived  are  then  used  in  deducing  the 
place  of  the  star  at  any  future  date  in  the  manner  indicated 
in  formulae  (8),  (9),  (5),  and  (6),  §§  41,  42.  For  a  full  discus- 
sion of  the  treatment  of  proper  motion,  from  the  astronomer's 
point  of  view,  with  the  refinements  necessary  when  reductions 
are  to  be  made  covering  very  long  periods  of  time,  see 
Doolittle's  Practical  Astronomy,  pp.  578-583,  and  Chauve- 
net's  Astronomy,  vol.  I.  pp.  620-623. 

A  concrete  idea  of  the  magnitude  of  the  proper  motion 
usually  found  may  be  gained  from  the  fact  that  in  the  Boss 
Catalogue  of  500  Stars  for  the  epoch  1875.0  there  are  only 
8  stars  out  of  the  500  for  which  the  annual  proper  motion  in 
declination  exceeds  o".5O.  In  367  cases  it  is  less  than  o".  10. 
Proper  motions  in  right  ascension  are  of  the  same  order  of 
magnitude, — keeping  in  mind,  of  course,  that  Is  of  right 
ascension  represents,  for  a  star  near  either  pole,  a  much 
smaller  displacement  upon  the  celestial  sphere  than  Is  for  a 
star  near  the  equator. 

45.  That  the  so-called  proper  motion  is  not  really  due  to 
an  erroneous  determination  of  the  precession,  is  put  in  evi- 
dence by  the  fact  that  the  various  proj>er  motions  for  different 
stars  do  not  show  the  systematic  relation  which  they  must 
necessarily  have  if  due  to  a  shifting  of  the  reference  circles. 
Precession  does  not  change  the  relative  positions  of  the  stars. 
Proper  motions  do.  To  what  are  the  proper  motions  due  ? 
rst.  If  they  are  due  to  a  motion  of  the  solar  system  as  a 
whole  through  space,  the  stars  which  are  ahead  in  the  direc- 
tion of  motion  must  seem  to  be  separating  in  all  directions 
from  the  point  toward  which  we  are  moving,  must  seem  to  be 
going  backward  at  the  sides,  and  apparently  closing  together 
behind  us, — just  as  points  of  the  landscape  seem  to  a  traveller 


50  GEODETIC  ASTRONOMY.  §  46. 

to  move.  2d.  If  the  proper  motions  are  due  to  actual  motions 
of  the  stars  themselves,  acting  as  entirely  independent 
bodies,  the  proper  motions  should  seem  to  be  without  any 
relation  to  each  other.  3d.  If,  on  the  other  hand,  they  are 
due  to  actual  motions  of  the  stars,  which  are  not,  however, 
independent,  one  would  expect  to  find  laws  connecting  the 
proper  motions, — laws,  however,  which  would  differ  from 
those  called  for  by  the  last  supposition  above.  A  close  study 
of  the  proper  motions  seems  to  indicate  that  there  is  some 
truth  in  each  of  the  three  suppositions. 

Computations  based  upon  hundreds  of  observed  proper 
motions,  made  by  different  astronomers  at  various  times,  have 
all  agreed,  in  a  general  way,  in  indicating  that  there  is  a  slow 
motion  of  the  solar  system  as  a  whole  through  space  toward 
a  point  in  the  neighborhood  of  a  =  i/h,  d  =  -j-  35°.  For 
details  in  regard  to  the  computations,  see  Chauvenet's  As- 
tronomy, vol.  I.  pp.  703-708.  In  regard  to  the  third  sup- 
position, it  may  be  noted  that  in  a  few  rare  cases  of  double 
stars,  two  stars  apparently  very  near  to  each  other,  the 
observed  proper  motions  indicate  that  the  two  revolve  about 
some  common  centre — are  linked  together  by  gravitation. 
But  though  some  laws  connecting  the  various  proper  motions 
have  been  thus  discovered,  the  salient  fact  to  keep  in  mind  is 
that  the  second  supposition  is  very  largely  true,— that  the 
discovered  laws  only  account  for  an  extremely  small  fraction 
of  the  actually  observed  proper  motions. 

Reduction  from  Mean  to  Apparent  Place. 

46.  To  reduce  from  the  mean  place  at  the  beginning  of 
the  year  to  the  apparent  place  at  a  given  date,  it  is  necessary 
to  reduce  the  mean  place  up  to  date,  and  then  apply  to  that 
result  the  effect  of  nutation  and  aberration  at  that  date. 


§47-  MEAN   TO   APPARENT  PLACE.  5  1 

This  computation,  if  made  directly  from  the  known  laws  of 
nutation  and  aberration,  is  very  laborious.* 

But  such  a  direct  computation  is  not  necessary.  This  is 
again  one  of  the  cases  in  which  it  is  advisable  for  the  engineer 
simply  to  accept  the  results  of  the  astronomer's  investigations 
in  the  convenient  form  in  which  they  are  given  in  the 
Ephemeris,  without  going  through  all  the  details  of  the 
derivation  of  those  results. 

47.  Suffice  it  to  say  that  this  reduction  has  been  put  in 
the  following  convenient  form  : 


sin  (G  +  a0)  tan  tfo 
sin  C^+  <*<>)  sec  <$o  •   •   •  (m  time);  (10) 


=    t--r/«£-  cos 

-f-  h  cos  (H  '  +  <x0)  sin  #0  -f-  i  cos  #0  .  (in  arc);          (i  i) 

in  which  a  and  d  are  the  required  apparent  right  ascension 
and  declination  at  some  stated  time;  a0  and  <50  are  the  mean 
right  ascension  and  declination  at  the  beginning  of  that 
fictitious  year;  r  is  the  elapsed  portion  of  the  fictitious  year 
expressed  in  units  of  one  year;  //  and  //  are  the  annual 
proper  motions  in  right  ascension  and  declination;  and/,  G, 
H,  g,  h,  and  i  are  quantities  called  independent  star-numbers^ 
which  are  functions  of  the  time  only,  and  are  given  in  the 
Ephemeris  for  every  Washington  mean  midnight  during  the 
year  (pp.  285-292  of  the  volume  for  1892).  r  expressed  in 
units  of  a  year  is  also  given  in  the  Ephemeris  on  the  same 
pages.  The  values  of  these  constants  may  be  derived  for  the 
exact  instant  at  which  they  are  required,  with  sufficient 

accuracy,  by  interpolations  along  chords. 

\ 

*  For  an  exhibit  of  the  formulae  for  the  computation  if  made  thus,  see 
Doolittle's  Practical  Astronomy,  p.  610. 


$2  GEODETIC  ASTRONOMY.  §  48. 

48.  The  computation  of  the  apparent  place  of  //  Hercules 
at  its  transit  at  St.  Louis,  Mo.,  July  16,  1892,  as  proposed 
in  §  40  and  partially  carried  out  in  §  43,  may  now  be  con- 
tinued, as  follows.  From  §  43  : 

a0  =  I7h  42m  i3"-9  =  265°  31'  (to  nearest  minute). 
So  =  27°  47'  02".37. 

St.  Louis  is  west  of  Washington 53™ 

St.  Louis  sidereal  time  of  transit  (same  as  a) 17^  42 

Washington  sidereal  time  (to  nearest  minute) 18    35 

Sidereal  time  of  mean  midnight  (at  end  of  the  civil  day,  July  16) 
by  interpolation  between  sidereal  times  of  mean  noon  as  given 

in  Ephemeris,  p.  381,  for  July  16  and  17  (to  nearest  minute) 19   42 

Hence  the  sidereal  interval  before  Washington  midnight  for  the 

stated  time  is i    07 

This  interval  is,  with  sufficient  accuracy  for  the  purpose  of  interpolation 

jh  Ojm 

of  the  star-numbers,  —  =  0.05  dav. 

24h 

The  Ephemeris,  p.  289,  gives  directly  the  following  values: 
July  15,  Washington  mean  midnight: 

r  f  G  H  log?  log  A  log/ 

0.54     +  i9.oo    315°  26'     157°  43'     +0.9607     +1.3048     +0.5212 

July  16,  Washington  mean  midnight: 

0.54     +  I8.oo     315°  41'     156°  49'     +0.9620     +1.3043     +0.5317 

The  signs  attached  to  log  g,  log  h,  log  i  in  the  Ephemeris 
are  the  signs  of  g,  h,  and  z,  and  not  signs  applying  to  their 
logarithms,  as  might  naturally  be  supposed  from  the  way  in 
which  they  are  printed. 

For  the  stated  time,  0.05  day  before  Washington  mean 
midnight  of  July  i6th,  following  the  order  indicated  by 
formula  (10), 

a0  =  I7b     42m     I38.86 

/=  +  i. oo 

(jj.  not  being  given)     r/i  =  o.oo 

log  y1^  =  8.8239 
=  0.9619 


§48.  MEAN   TO   APPARENT  PLACE.  53 

G  =  315°  40'.  (G  -f  <*o)  =  221°  13',  log  sin  (G  -f  <*<>)  =  9.8185* 

log  tan  S0  =  9.7217 


log  T5«?  sin  (G  +  <*o)  tan  S0  =  9.3263* 

y1^  sin  (G  -f  <*o)  tan  £0  =  —0.21 

log  iV  =  8.8239 
log  h    —  1.3043 

H  —  156°  52',  (#"  +  a0)  =  62°  25',  log    sin  (ff+  a0)  =  9-9475 

log  sec  50  =  0.0532 


log  •&&  sin  (•#"  +  <*<>)  sec  d0  =  0.1290 
/fc  sin   H--  «o    sec  S0 


a,   at  i6h  iom  14*.!.     St.  Louis  Sidereal  Time,  July  16,  1892  =17''  42™  i68.oo 
The  computation  for  5,  following  the  order  of  (n)  is 

d0  =  27°  47'  02".37 
TJJ.'  =  (o.54)(—  o".76),    [nf  is  given  =  —  o".76  in  Boss'  list]  =  —  o  .41 

log  S  —  o-9OI9 
log  cos  (G  -f  ao)  =  9.8766* 

log  g  cos  (G  -f  tfo)  =  0.8382 

£-cos(£+  o-0)  =  -  6  .89 

log  h  =  1.3043 

log  cos  (/f  +  <*o)  =  9.6661 

log  sin  <50  =  9.6685 


log  h  cos  (If-}-  cto)  sin  d0  •=.  o  6384 

h  cos  (//+  a0)  sin  dQ  =  -j-  4  .35 

log  f  =  0.5363 
log  cos  d0  =  9.9468 


log  i  cos  50  =  0.4831 

*  cos  d0  =4-3  .04 


6,    at  i6h  iom  i4«.i.    St.  Louis  Sidereal  Time,  July  16,  1892  =  27°  47'  O2".46 

The  above  example  shows  how  far  out  the  computation 
needs  to  be  carried.  Where  many  star  places  are  to  be  com- 
puted, the  computation  is  materially  shortened  by  using 
printed  blank  forms  so  arranged  as  to  facilitate  the  work. 


54  GEODETIC  ASTRONOMY.         .  §  49. 

Especially  convenient  forms  of  that  nature  are  in  use  in  the 
Coast  and  Geodetic  Survey. 

49.  In  computing  a  number  of  stars  on  a  single  night — 
which  will  usually  be  the  case  in  dealing  with  latitudes 
observed  with  a  zenith  telescope — considerable  time  will  be 
saved  at  an  exceedingly  small  sacrifice  of  accuracy  by  the 
following  procedure.  First  interpolate  the  values  of  the 
independent  star-numbers  for  every  whole  hour  from  Wash- 
ington mean  midnight  for  the  period  over  which  the  oberva- 
tion  extends.  Then  for  each  star  use  the  interpolated  value 
of  each  star-number  for  the  nearest  hour  as  interpolated, 
instead  of  making  a  special  interpolation  for  each. 

For  an  account  of  the  method  of  computation  of  the 
independent  star-numbers,  and  the  method  of  computing  star 
places  by  the  use  of  the  Besselian  star-numbers,  see  Doolittle's 
Practical  Astronomy,  pp.  609-617;  Chauvenet's  Astronomy, 
vol.  I.  pp.  645-651 ;  and  the  Ephemeris,  pp.  280-284  (°f  tne 
volume  for  1892).  The  Besselian  star-numbers  are  not  ordi- 
narily so  convenient  for  the  engineer  as  the  independent  star- 
numbers. 

If  one  has  a  great  number  of  star  places  to  compute, 
under  certain  conditions,  the  work  may  be  abridged  somewhat 
by  using  differential  and  graphic  methods.  For  the  details 
of  a  differential  method  which  reduces  the  labor  of  computa- 
tion about  one-half  in  case  the  place  of  each  star  is  to  be 
computed  on  three  or  more  nights,  see  Coast  and  Geodetic 
Survey  Report,  1888,  pp.  465-470.  A  somewhat  similar 
method  to  be  used  when  the  places  are  to  be  computed  for  a 
few  stars  on  many  nights  will  be  found  in  the  Coast  and 
Geodetic  Survey  Report  for  1892,  Part  II,  pp.  73-75.  For 
a  graphic  method  of  reducing  from  the  mean  to  the  apparent 
place  in  declination,  see  Coast  and  Geodetic  Survey  Report, 
1895,  pp.  371-380. 


§  50.  QUESTIONS  AND   EXAMPLES.  55 

QUESTIONS   AND    EXAMPLES. 

50.  I.  At  a  certain  instant  in  the  forenoon  of  May  27, 
1892,  the  observed  hour-angle  of  the  Sun  at  Cornell  Observa- 
tory was  2h  30™  41s.  What  was  its  apparent  right  ascension 
and  declination  at  that  instant  ?  Cornell  is  2m  i6s  east  of 
Washington.  The  Ephemeris  for  1892  (p.  380)  gives  for 
Washington  apparent  noon  May  26th  a  =  4h  15™  47*.93, 
d  =  2i°  if  45". 3,  hourly  motion  in  right  ascension  =  -f- 
ios.i4i,  hourly  motion  in  declination  =-\-2$".i6;  and  for 
Washington  apparent  noon  May  2/th  a  =  4h  19™  5iB.57, 
6  =  21°  27'  38//.3,  hourly  motion  in  right  ascension  =  -f- 
l68. 160,  in  declination  =  -f-  24//-23- 

Ans.  By  interpolation  along  a  tangent  a  =  4h  19™  258.67, 
6  =  21°  26'  36".5. 

By  interpolation  along  a  parabola  a  =  4h  I9m  25s. 67, 
6  =  21°  26'  $6". 4. 

2.  What    was    the   apparent   declination   of    the    Sun   at 
7h  4ira  32s  A.M.   Greenwich  mean  time  on   Dec.  21,    1892  ? 
The  Ephemeris  for  1892  (p.  201)  gives  the  apparent  declina- 
tion of  the  Sun  at  Greenwich  mean  noon  Dec.  2ist  =  —  23° 
27'   i8".6,  and  its  hourly  motion  =  +  o".  18;  and  for  Dec. 
20th    <$  =  —  23°    27'    08". 7,    with    an    hourly   motion  =  — 
i".oo.  Ans.    —  23°  27'  i8".9. 

3.  Work  the  preceding  problem,  as  a  check,  from  the  fol- 
lowing data  from  the   Ephemeris   (p.   384):    Declination   of 
Sun  at  Washington  mean  noon  Dec.  2ist  =  —  23°  27'  17". o, 
hourly  motion  =  -\-  o".43.      Hourly  motion  for  Washington 
mean  noon  Dec.  2Oth  =  —  o".75.     Washington  is  5h  8m  I28 
west  of  Greenwich.  Ans.    —  23°  27'  i8".9. 

4.  At  a  station   2h   58™  west  of  Washington    the  south 
zenith    distance   of  Jupiter,    at   the   instant   of  its   meridian 
transit  on  July  16,  1892,  was  observed  to  be  29°  22'  17". 4. 


$6  GEODETIC  ASTRONOMY.  §  50. 

What  was  the  latitude  of  the  station  ?  The  apparent 
declination  of  Jupiter  at  its  meridian  transit  at  Washington  is 
given  in  the  Ephemeris  (p.  404)  as  follows:  July  I5th  -f-  7° 
56'  u".8,  July  1 6th  +  7°  57'  53".8,  and  July  i;th  +  7°  59' 
32//.o.  ^»,y.  37°  21'  23".5. 

5.  What  was  the  right  ascension  and  declination  of  the 
Moon  at  8h  30™  09"  P.M.  local  mean  time  at  Cornell  April  8, 
1892  ?     Cornell   is    5h    5m   56s  west   of    Greenwich.      In    the 
Ephemeris   (p.    62)   the   position    of  the   Moon  is  given   for 
2h  A.M.  Greenwich  mean  time  April  9th,  a  =  iih  33™  438.9i, 
3  =  -f-  7°   34'  02". 2;  "  difference    for    i    minute"    in    right 
ascension  =  +  Is- 7980;  "  difference  for  I   minute  "  in  decli- 
nation =  —  1 3".  192.     The  differences  for   I  minute  in  right 
ascension  and  declination  respectively  at  ih  A.M.  are  -f-  I8.8oo3 
and  —  13".  167. 

Ans.   By  interpolation  along  a  tangent  a  =  nh  33™  oos.9i, 

S  =  7°  39'  i7".7- 

By  interpolation  along  a  parabola  a  =  i  ih  33m  oos.9O, 
6=7°  39'  I7".6. 

6.  What  was  the  apparent  right  ascension  of  the  star  A 
Aquarii  at  transit  at  Mount  Hamilton,  Cal.,  Sept.  i,  1892  ? 
For   upper  transit   at   Washington   (Ephemeris,    p.    362)   on 

August  27th  a  =  22h  46™  6is.5i,  and  —=-  =  +  os.  10  per  ten 

days.     Also  for  Sept.  6th  a  —  22h  46™  6i8.66,  and  -^  —  + 

O8.O5  per  ten  days.  The  longitude  of  Mount  Hamilton  is 
2h  58m  22s  west  of  Washington. 

Ans.   By  interpolation  along  a  chord  a  —  22b  47™  ois.63. 

By  interpolation  along  a  tangent  from  Sept.  6th  a  '=  22h 
47m  oi8.64. 

By  interpolation  along  a  parabola  a  =  22h  47™  oi8.63. 

7.  For  star  BAG  5706  *I875  =  i6h  50™  4?. 9,   dl875  =  46° 


§  SO-  QUESTIONS  AND   EXAMPLES.  57 

44'  3 1  ".22.  Its  annual  variation  in  right  ascension  for  that 
date  =  -f-  Is. 72 1,  and  in  declination  (including  proper  motion) 
=  —  6''. 0105.  What  was  its  mean  declination  for  1895.0  ? 

Ans.   <?I89S  =  46°  42'  3ix/.49- 

8.  For    the    star    TJ   Geminorum    «I875  =  6h   07™    2O8.o, 

— rp  =  +  3S.622  per  year,  #l875  =  22°  32'  27".  18,  annual  pre- 
cession in  declination  =  —  o".64i5,  annual  proper  motion  in 
declination  =  —  o//.oi6i.  What  is  its  mean  declination  for 
1892.0?  Ans.  dl892  =  22°  32'  15". 24. 

9.  For  the  star  BAG  7440,  al8t)2  =  2ih  I9m  39%  tfl892  =  — 
4°  01'  1 1 ".30,  and  annual  proper  motion  in  declination  —  — 
o//.o68.     For  the  star  BAG  7482  alS92  —  2ih  25™  41%  tfI&92  = 
66°  20'  i6".oo,  and  annual  proper  motion  =  —  0^.04.2.    What 
was  the  apparent  declination  of  each  of  these  stars  at  transit 
on  August  9th  and  i6th  at  San  Bernardino  Ranch,  Arizona, 
2h  09™  west  of  Washington  ?     The  Ephemeris  (pp.  289,  290) 
gives  the  following  data  for  Washington  mean  midnight: 

Aug.  9.  Aug.  10.  Aug.  16.  Aug.  17. 

r 0.61  0.61  0.63  0.63 

G 319°  33'  319°  29'  320°  38'  320°  54' 

H 134°  38'  133°  30'  127°  38'  126°  39' 

logg 1.0309  1.0328  1.0418  1.0449 

log  h 1.2907  1.2900  1.2863  1.2857 

log  i 0.7815  0.7879  0.8226  0.8276 

The  sidereal  time  of  mean  midnight   at  Washington   on 
Aug.  9th  was  2ih  I7m,  and  on  Aug.  i6th,  2ih  44™. 

Ans.  BAG  7440,  Aug.  9th,  d  —  —  4°  01'  O3//.43. 

Aug.  i6th,  d  =  —  4°  oix  02//.75. 

BAG  7482,  Aug.  9th,  6  —  66°  20'  i8".63. 

Aug.  i6th,  d  =  66°  207  2i".26. 

10.  Justify  the  half  square  in  the  last  term  of  formula  (9), 

73   » 

§  42.     That  is,  show  that  (9)  is  exact  if  —jjr  =  o. 


58  GEODETIC  ASTRONOMY.  §  50. 

11.  Show   from   formula  (5),    §  41,    that   —~  cannot  be 

negative   for    any   star   unless  its  declination  is   quite   large 
(near  90°).     What  other  condition  must  also  be  fulfilled  ? 

12.  Draw   and    explain    the    diagram    called    for  in   the 
parenthesis  in  §  38,  showing  the  geometrical  relation  between 
the  aberration,  velocity  of  light,  and  velocity  and  direction 
of  motion  of  the  observer. 

13.  Look  in  the  Ephemeris  and  see  whether  r  as  given 
with  the  independent  star-numbers  for  Jan.  o  of  the  current 
year  is  zero.      If  not,  why  not  ?     At  what  time  during  this 
year  was  it  exactly  zero  ? 


§52.  THE   SEXTANT.  59 


CHAPTER    III. 
THE   SEXTANT. 

51.  The  sextant  is  an  instrument  for  measuring  angles; 
especially  useful  at  sea  and  on  exploratory  surveys  because  of 
its  lightness  and  portability,  and  because  it  requires  no  fixed 
support.      It    also    commends    itself    in    certain    other    cases 
because    results   of  a  sufficient   degree   of    accuracy  can   be 
obtained  with  it  more  conveniently  than  with  larger  instru- 
ments with  fixed  supports  which  might  otherwise  be  employed. 
It  is  principally  used  in  astronomy  for  the  determination  of 
local  time  and  of  latitude  at  sea,  and  on  explorations  by  land. 
It  is   also  used  extensively  in  hydrographic  surveying  for  the 
measurement   of  horizontal  angles  serving  to  locate  sound- 
ings.    With  it  an  angle  may  be  measured  from  the  deck  of  a 
rolling  vessel  where    an  engineer's    transit  or   a    theodolite 
would  be  unavailable. 

Description  of  the  Sextant. 

52.  A  view  of  a  sextant  is  shown  in  Fig.  3.      The  main 
frame  ABC  carries  a  graduated  arc,  DE,  of  which  the  center 
is  at  F\  it  carries  a  bearing  at  F  which  receives  the  axis  (per- 
pendicular to  the  plane  of  the  frame)  about  which  the  arm 
GF  swings;  a  ring  secured  to  the  frame  at  H  into  which  the 
telescope  /  is  screwed  and  held  in  a  fixed  position  relatively 
to  the  frame;  a  plane  mirror  aty,  called  the  horizon-glass  y 
which  is  fixed  to  the  frame  in  a  plane  perpendicular  to  it ;  and 
certain  colored  glasses  at  K  and  L,  which  may  be  used  to 


60  GEODETIC  ASTRONOMY.  §  53. 

absorb  some  of  the  light  when  observing  the  Sun  (or  the 
Moon  with  a  star),  so  that  the  images  seen  in  the  telescope 
may  not  be  too  bright  for  comfort.  N  is  the  handle  by  which 
the  sextant  is  held  in  the  observer's  right  hand.  The  arc 
DE  is  graduated  to  five-minute  spaces,  but  has  the  gradua- 
tion marked  upon  it  as  if  each  space  were  TEN  minutes.  The 
arm  GF  carries  a  vernier  at  G,  which  reads  against  the  arc 
DE  to  five  seconds  (real).  It  is  marked,  however,  as  if  it 
read  to  ten  seconds.  Any  reading  on  the  arc  DE  made  by 
means  of  the  vernier  indicates,  therefore,  twice  the  angle 
between  that  position  of  the  arm  GF  and  the  position  corre- 
sponding to  the  zero  reading.  M  is  a  small  glass  used  in 
reading  the  vernier. 

The  arm  GF  also  carries  at  F  a  plane  mirror,  called  the 
index-glass,  which  is  perpendicular  to  the  plane  of  the  frame 
for  any  position  of  the  arm.  The  horizon-glassy  has  only 
that  half  of  its  surface  which  is  nearest  the  sextant  frame 
silvered.  The  other  half  is  merely  a  plane  clear  glass,  or  is 
cut  away  entirely.  The  telescope  ./  may  be  adjusted  to 
such  a  distance  from  the  sextant  frame  that  the  edge  of  the 
silvering  of  mirror  J  is  in  the  axis  of  the  telescope  produced. 
The  observer  thus  sees  at  the  same  time  both  the  images 
reflected  from  the  silvered  surface  and  whatever  may  be  in 
the  line  of  sight  of  the  telescopt  beyond  the  mirror. 

The  Principle  of  the  Sextant. 

53.  The  principle  underlying  sextant  observations  is  indi- 
cated by  Figs.  4  and  5.  Suppose  the  sextant  to  be  in  perfect 
adjustment.  Let  OP,  Fig.  4,  be  a  ray  of  light,  from  a  dis- 
tant object,  which  passes  through  the  unsilvered  portion  of 
mirror  J,  without  change  of  direction,  into  the  telescope  /, 
parallel  to  its  axis.  Let  QR  be  a  ray  of  light,  parallel  to  OP, 
which  strikes  the  index-glass  F.  The  positions  of  /''relative  to 


§  53-  THE   SEXTANT  PRINCIPLE.  6 1 

the  arm  FG,  and  of  J  relative  to  the  telescope,  are  such  that 
if  the  reading  of  the  arc  taken  from  vernier  G  is  zero  the  ray 
QR  will  be  reflected  from  F  along  the  line  FJt  and  reflected 
again  from  the  silvered  portion  of  J  along  a  line  parallel  to 
OP  into  the  telescope.  The  image  of  the  object  from  which 
QR  came  will  therefore  be  seen  in  the  telescope  in  coincidence 
with  the  image  of  the  object  from  which  OP  came.  To 
secure  the  above  result  mirrors  F  and  J  must,  for  this  zero 
position  of  the  arm,  be  parallel,  and  the  perpendiculars  to  the 
mirrors,  FS  and  JT,  must  bisect  the  angles  QFJ  and  FJP. 
Note  that  the  arc  reading  is  the  same  as  the  angle  (zero  in 
this  case)  between  the  two  rays  of  light  QR  and  OP  which 
eventually  enter  the  telescope  as  parallel  rays. 

In  Fig.  5  let  OP  be  as  before;  but  let  Q Rr  be  a  ray  of 
light  at  an  angle  ft  with  the  ray  OP  (or  with  the  ray  QR  of 
the  preceding  paragraph)  and  striking  the  index-glass.  Evi- 
dently QR'  will  not  be  reflected  to  /  from  F  unless  F  is  first 

8 
rotated  through  the  angle  -  by  moving  the  arm  FG  to  the 

position  FG'.  In  this  position  of  the  mirror  F  the  perpen- 
dicular FS'  will  bisect  the  angle  Q'FJ.  The  angle  GFG'  will 

n 

be  -,  but  on  account  of  the  peculiar  graduation  of  the  arc  as 

indicated  in  §  52  the  reading  of  the  arc  will  be  ft.  The  ray 
FJ  will  evidently  be  reflected  into  the  telescope,  as  before, 
along  a  line  parallel  to  OP,  and  the  object  from  which  QR 
came  will  be  seen  in  the  telescope  apparently  in  coincidence 
with  the  object  from  which  OP  came.  Note  that  here,  as 
before,  the  reading  of  the  arc  is  the  angle  between  the  two 
rays  of  light  QR'  and  OP. 

So  for  any  case,  if  the  sextant  is  in  perfect  adjustment, 
the  reading  of  the  arc  is  the  angle  between  two  rays  of  light, 
one  coming  to  the  index-glass  and  the  other  through  the  un- 


62  GEODETIC  ASTRONOMY.  §  55. 

silvered  portion  of  the  horizon-glass,  which  finally  reach  the 
telescope  as  parallel  rays  and  produce  coincident  images. 
When  the  images  of  two  celestial  objects,  or  any  two  objects 
at  a  great  distance  from  the  observer,  are  made  to  apparently 
coincide  in  the  telescope  the  arc  reading  is  the  angle  at  the 
eye  between  the  two  objects,  measured  in  the  plane  defined 
by  the  two  objects  and  the  eye.  This  plane  may  happen  to 
be  in  a  horizontal,  a  vertical,  or  an  oblique  position.  If  the 
two  objects  observed  are  at  a  comparatively  short  distance  it 
may  be  necessary  to  take  account  of  the  fact  that  the  angle 
indicated  by  the  arc  is  the  angle  between  the  two  objects 
from  the  point  U  in  which  Q ' R!  produced  intersects  OP,  and 
not  the  angle  at  the  eye.  The  difference  between  these  two 
angles  is  called  the  sextant  parallax. 

Adjustments  of  the  Sextant. 

54.  To  make  the  index-glass  perpendicular  to  the  plane  of 
the  sextant  * — Place  the  vernier  near  the  middle  of  the  arc. 
Hold  the  instrument  with  the  arc  away  from  you,  and  look 
obliquely  into  the  index-glass  in  such  a  way  as  to  see  a  por- 
tion of  the  arc  both  directly  and  by  reflection  at  the  same 
time.     If  the  direct  and  reflected  portions  appear  to  form  one 
continuous  arc  the  adjustment  is  perfect.      If  riot,  the  inclina- 
tion of  the  glass  to  the  plane  of  the  sextant  must  be  changed 
by  whatever  means  have  been  provided  on  that  particular 
instrument.     This  adjustment  once  carefully  made  will  not 
require  frequent  attention;  for  this  reason  some  makers  do 
not  provide  a  convenient  means  of  making  it. 

55.  To  make  the  horizon-glass  perpendicular  to  the  plane  of 
the  sextant. — Having  first  adjusted  the  index-glass,  point  the 

*  By  " plane  of  the  sextant"  is  meant  the  plane  of  the  graduated  arc, 
to  which  the  axis  about  which  the  arm  rotates  is  necessarily  perpendic- 
ular. 


§  5&  ADJUSTMENTS.  63 

telescope  to  any  well-defined  object  and  move  the  vernier 
slowly  back  and  forth  past  the  zero.  The  reflected  image 
will  be  seen  moving  back  and  forth  past  the  direct  image. 
If  in  passing  it  coincides  with  the  direct  image,  the  adjust- 
ment is  perfect.  If  not,  the  correction  must  be  made  by  use 
of  the  screws  provided  for  that  purpose  at  the  back  of  the 
horizon-glass.  With  most  sextants  this  adjustment  must  be 
inspected  frequently. 

A  star  is  the  best  object  for  this  purpose.  The  Sun  may 
also  be  employed.  The  accuracy  will  be  increased  by  making 
the  two  images  of  the  Sun  appear  of  different  colors  by  use 
of  the  colored  glass  shades.  The  sea  horizon  may  be  used  by 
holding  the  plane  of  the  sextant  horizontal  and  keeping  the 
arc  reading  nearly  zero. 

56.  To  make  the  axis  of  the  telescope  parallel  to  the  plane 
of  the  sextant. — Rotate  the  eye  end  of  the  telescope  until 
two  of  the  four  dark  lines  seen  in  the  telescope  are  parallel  to 
the  plane  of  the  sextant.*  Point  the  telescope  to  one  of  two 
objects  at  an  angle  of  90°  or  more  apart.  Bring  the  reflected 
image  of  the  second  object  into  contact  with  the  image  of  the 
first  at  that  one  of  the  parallel  lines  which  is  apparently 
nearest  the  sextant  frame.  This  may  be  done  by  rotating 
the  sextant  about  the  telescope  as  an  axis  until  it  is  in  the 
plane  of  the  two  objects  and  the  eye,  and  then  bringing  the 
vernier  to  the  proper  reading  by  trial.  Now  move  the  instru- 
ment, without  changing  the  vernier  reading,  so  that  the  two 
images  are  upon  the  other  of  the  two  parallel  lines.  If  the 
contact  is  still  perfect  no  adjustment  is  required.  Otherwise 
the  ring  into  which  the  telescope  is  screwed  must  be  adjusted 
to  change  the  inclination  of  the  telescope  to  the  sextant  plane 
until  the  above  test  fails  to  detect  any  error. 

*  These  lines  are  placed  by  the  instrument-maker  in  symmetrical  posi- 
tions on  each  side  of  the  middle  of  the  field  of  the  telescope.  All  observa- 
tions are  to  be  made  at  about  the  middle  of  the  space  defined  by  them. 


64  GEODETIC  ASTRONOMY.  §  5/. 

This  adjustment  may  also  be  made  as  follows:  Place  the 
sextant  face  upward  on  a  table  or  other  firm  horizontal  sup- 
port. Sight  across  in  the  plane  of  the  graduation,  or  a 
parallel  plane,  and  mark  a  point  at  that  height  and  in  line 
with  the  telescope,  upon  a  wall  distant  fifteen  feet  or  more. 
Measure  upward  from  this  point  a  distance  equal  to  the 
measured  distance  at  the  sextant  from  the  sight  plane  just 
used  to  the  axis  of  the  telescope,  and  mark  this  second  point. 
The  ring  carrying  the  telescope  must  now  be  moved,  if  neces- 
sary, in  such  a  w.iy  as  to  change  the  inclination  of  the  tele- 
scope until  this  second  point  is  exactly  in  the  centre  of  the 
field  of  the  telescope. 

It  should  be  noted  that  when  this  adjustment  has  been 
accurately  made  a  contact  made  upon  one  of  the  side  lines 
will  not  necessarily  be  perfect  when  shifted  to  the  middle  of 
the  field.  The  reading  of  the  arc  is  slightly  less  for  a  contact 
made  at  the  middle  of  the  field  than  for  one  made  on  either 
side  when  all  adjustments  are  perfect.* 

This  adjustment  will  usually  remain  sensibly  perfect  for  a 
long  period. 

57.  To  make  the  index  error  zero. — The  reading  of  the 
arc  when  the  direct  and  reflected  images  of  the  same  point  f 
are  made  to  coincide  is  called  the  index  error  of  the  sextant. 
The  negative  of  the  index  error  is  the  index  correction^  which 
evidently  must  be  applied  to  every  reading.  To  make  the 
index  error  zero  the  horizon-glass  may  be  rotated  about  a  line 
perpendicular  to  the  plane  of  the  sextant.  Screws  for  pro- 
ducing this  rotation  are  often,  though  not  always,  provided 


*  For  a  detailed  statement  of  the  theory  of  the  errors  arising  from  non- 
parallelism  of  telescope  to  the  plane  of  the  sextant,  see  Chauvenet's 
Astronomy,  vol.  II,  pp.  112-114. 

f  Provided,  of  course,  that  the  point  is  so  distant  that  the  sextant 
parallax  (§53)  may  be  neglected. 


§58.  DIRECTIONS  FOR   OBSERVING.  65 

by  the  sextant-maker.  Since  this  adjustment  cannot  ordi- 
narily be  depended  upon  to  remain  perfect  even  for  a  day,  it 
is  advisable  to  determine  the  index  error  at  the  time  of  each 
series  of  observations,  and  apply  the  derived  correction, 
instead  of  trying  to  keep  the  value  of  the  error  down  to  zero 
by  adjustment.  When  this  procedure  is  adopted  it  is  only 
necessary  to  make  the  adjustment  at  rare  intervals  when  the 
index  error  has  become  inconveniently  large.  The  method 
of  determining  the  index  error  will  be  found  in  §  62. 

Directions  for  Observing  the  Sun's  Altitude  with  a  Sextant  to 
Determine  the  Local  Time. 

58.  The  altitude  of  the  Sun  is  a  known  function  of  the 
latitude  of  the  station  of  observation,  of  the  declination  of  the 
Sun,  and  of  the  local  apparent  solar  time.  Hence  if  the  lati- 
tude and  declination  are  known,  and  the  altitude  is  measured, 
at  a  given  instant,  the  local  apparent  solar  time  may  be  com- 
puted. From  this  the  mean  solar  time  may  be  derived. 

In  determining  the  altitude  of  the  Sun  at  a  station  on  land 
the  artificial  horizon  must  be  used.  The  artificial  horizon  is 
a  shallow  rectangular  basin  filled  with  mercury,  molasses,  or 
oil,  protected  from  the  wind  by  a  roof  consisting  of  two  pieces 
of  plate  glass  held  in  a  suitable  mounting.  These  glass  plates 
each  have  faces  which  are  as  nearly  as  possible  plane  and 
parallel,  so  that  rays  of  light  may  pass  through  them  without 
unequal  change  of  direction. 

Let  MN,  Fig.  6,  represent  the  surface  of  the  mercury  in 
the  artificial  horizon.  MN  is  necessarily  a  horizontal  surface, 
that  is,  a  surface  which  is  perpendicular  at  every  point  to  the 
action  line  of  gravity  at  that  point.  A  ray  of  light  SB  from 
the  Sun  will  be  reflected  along  a  line  BA  in  the  same  vertical 
plane  with  SB,  and  such  that  the  angle  NBA  is  equal  to  the 
angle  MBS.  An  observer  at  A  will  see  the  reflected  image 


66  GEODETIC  ASTRONOMY.  §  58. 

along  the  line  AS" .  He  may  also  see  the  Sun  directly  along 
the  line  AS'.  The  distance  AB  being  very  small  as  compared 
with  the  distance  to  the  Sun,  S'A  and  SB  are  sensibly  parallel, 
and  the  angle  S'AS",  which  is  to  be  measured  with  the 
sextant,  is  evidently  the  double  altitude  of  the  Sun. 

Before  commencing  the  observations,  the  adjustments 
should  be  examined  and  corrected  if  necessary,  the  telescope 
should  be  carefully  focused  to  give  well-defined  images  of  the 
Sun,  and  such  colored  glasses  interposed  in  the  path  of  the 
light  that  the  two  images  of  the  Sun  will  be  of  about  the 
same  brightness,  and  dim  enough  so  that  continued  gazing  at 
them  will  not  fatigue  the  eye. 

To  begin  observations,  place  the  eye  in  such  a  position 
that  an  image  of  the  Sun  can  be  seen  reflected  from  the  arti- 
ficial horizon.  Without  moving  the  eye,  bring  the  telescope 
of  the  sextant  up  to  it,  and  point  upon  this  image.  Being 
careful  to  hold  the  plane  of  the  sextant  vertical,  swing  the 
vernier  slowly  back  and  forth  along  the  arc.  If  this  is  done 
with  sufficient  care,  a  second  image  of  the  Sun,  formed  by 
light  coming  to  the  telescope  by  way  of  the  index-glass,  will 
be  seen  in  the  telescope  when  the  vernier  is  near  the  reading 
of  the  arc  corresponding  to  the  double  altitude  of  the  Sun. 

For  convenience  let  the  two  images  of  the  Sun  be  called 
the  horizon  image  and  the  index-glass  image,  respectively. 

If  the  observer  has  not  had  sufficient  experience  to  handle 
the  sextant  with  facility,  it  will  be  well  for  him  at  this  point 
to  familiarize  himself  with  the  following  motions  and  their 
effects.  Rotate  the  sextant  about  the  telescope  as  an  axis: 
the  horizon  image  will  appear  to  remain  fixed  while  the  index- 
glass  image  will  appear  to  move  sidewise  horizontally.  Move 
the  vernier  slowly  along  the  arc,  keeping  the  sextant  frame 
and  telescope  fixed:  the  index-glass  image  will  appear  to 
move  vertically,  while  the  horizon  image  apparently  remains 


§59-  DIRECTIONS  FOR    OBSERVING.  67 

fixed.  Rotate  the  sextant  about  a  line  in  its  plane  perpen- 
dicular to  the  telescope  at  the  eye  end:  the  images  will 
appear  to  move  sidewise  without  change  of  relative  position. 
Rotate  the  sextant  about  a  line  perpendicular  to  its  plane  at 
the  eye :  and  the  images  will  appear  to  move  vertically  with- 
out change  of  relative  position.  Move  the  eye  horizontally 
in  the  vertical  plane  passing  through  the  artificial  horizon : 
and  the  horizon  image  will  appear  to  be  cut  off  by  a  straight 
line  on  the  upper  edge,  or  lower,  as  the  eye  is  moved  forward 
or  backward.  Similarly,  if  the  eye  is  moved  sidewise,  the 
horizon  image  will  be  seen  partially  cut  off  by  the  side  of  the 
artificial  horizon  on  one  side  or  the  other,  as  the  case  may  be. 
The  effects  of  these  various  movements  of  the  sextant  have 
been  commented  upon  because  the  ease  and  rapidity  with 
which  one  can  use  the  sextant  depends  largely  upon  having 
accurate  conceptions  of  these  effects,  as  well  as  upon  manual 
skill.  To  secure  steadiness  of  the  sextant,  it  is  well,  in  addi- 
tion to  holding  it  by  the  handle  in  the  right  hand,  to  rest  the 
lower  edge  of  the  arc  upon  the  fingers  and  thumb  of  the  left 
hand.  Care  must  be  taken,  however,  not  to  touch  the 
graduations  at  any  time;  nor  to  touch  any  part  of  the  vernier 
arm,  or  of  the  clamp  and  screws  attached  to  it,  at  the  instant 
when  an  observation  is  made. 

59.  The  observer  having  secured  control  of  the  images, 
let  them  be  placed  so  as  to  be  near  together,  one  above  the 
other,  and  approaching  each  other,  let  us  say.  For  the 
images  will,  in  general,  be  moving  relatively  to  each  other, 
since  the  altitude  of  the  Sun  is  continually  changing.  Clamp 
the  vernier  in  this  position.  Pick  up  the  beat  of  the  chro- 
nometer.* Then  watch  the  approaching  images,  keeping 
their  adjacent  portions  about  in  the  middle  of  the  field  of  the 

*  See  §  60. 


68  GEODETIC  ASTRONOMY.  §  60. 

telescope,  and  carefully  keeping  one  image  vertically  above 
the  other.  Note  the  exact  time  by  the  chronometer  when 
the  two  images  are  first  tangent  to  each  other.  Observe  and 
record  the  corresponding  reading  of  the  vernier.  Unclamp 
the  vernier  and  place  the  images  in  about  the  same  relative 
position  as  before,  and  repeat  the  process  until  six  readings 
(say)  of  time  and  the  corresponding  angle  have  been  made. 
Then  repeat  the  whole  process,  with  the  difference  that  now 
the  images  are  slightly  overlapped  at  first  and  allowed  to 
separate,  the  instant  when  the  tangency  of  the  images  takes 
place  being  noted.  To  complete  the  observation  of  the  Sun's 
altitude  it  now  remains  to  determine  the  index  error  (see 

§62). 

When  a  tangency  is  observed  with  images  approaching, 
the  noted  time  is  too  late  unless  the  observer  has  accurately 
kept  the  images  in  the  same  vertical  plane.  The  reverse  is 
true  of  an  observation  made  upon  separating  images.  To 
guard  against  an  error  in  locating  the  vertical  plane,  it  is  well 
to  continually  rotate  the  sextant  very  slightly  back  and  forth 
around  the  telescope  as  an  axis  so -as  to  be  certain  to  secure 
the  first,  or  last,  tangency,  as  the  case  may  be. 

60.  To  pick  up  the  beat  of  the  chronometer,  first  look  at 
some  second-mark  two  seconds  or  more  ahead  of  the  seconds 
hand.  Fix  the  name  of  that  second  in  mind  as  the  seconds 
hand  approaches  it.  Name  it  exactly  with  the  tick  at  which 
the  seconds  hand  reaches  it,  keeping  the  rhythm  of  the 
chronometer  beat.  Count  it  either  aloud,  in  a  whisper,  or 
mentally.  In  counting  it  will  be  found  easier  to  keep  the 
rhythm  if  the  names  of  the  numerals  are  elided  in  such  a  way 
as  to  leave  but  a  single  staccato  syllable  in  each.  The  half- 
second  beat  should  be  marked  by  the  word  "half"  thus: 
one,  half;  two,  half;  three.  .  .;  twenty r,  half;  twenty-*?;^, 
half;  twenty-/ie/0,  .  .  . ;  and  so  on.  With  practice,  an  observer 


§  6l.  DIRECTIONS  FOR   OBSERVING.  69 

can  carry  the  count  of  the  beat  for  an  indefinite  period  with- 
out looking  at  the  chronometer  face,  provided  he  can  hear 
the  tick.  If  he  becomes  expert,  he  will  even  be  able  to  carry 
the  count  for  a  half-minute  or  more  during  which  he  has  not 
even  heard  the  tick.  When  an  observation  is  made  of  the 
time  of  tangency  of  two  images,  or  of  any  other  visual  event, 
the  eye  observes  the  event  and  the  chronometer  is  read  by 
ear  at  the  same  instant.  It  is  conducive  to  accuracy  for  the 
observer  to  acquire  the  habit  of  deciding  definitely,  at  once, 
without  hesitation,  upon  the  second  and  fraction  as  soon  as 
he  has  seen  the  event.  He  who  hesitates  is  inaccurate. 

The  observation  of  time  may  be  made  by  the  observer  at 
the  sextant  calling  "  tip,"  at  the  instant  of  tangency,  to  an 
assistant  who  reads  the  face  of  the  chronometer  by  eye. 
This  is  an  easier  process,  but  is  also  a  much  less  accurate 
process  than  the  one  described  above.  The  nerve  times  (or 
intervals  of  time  required  for  the  nerves  concerned  to  perform 
their  functions),  and  errors  of  judgment,  of  two  men  instead 
of  one,  enter  into  the  result.  Moreover,  the  assistant  at  the 
chronometer  is  observing  an  event  which  comes  upon  him 
suddenly  instead  of  one  of  which  he  sees  the  gradual  approach. 
If,  however,  an  ordinary  watch  is  used  instead  of  a  chronom- 
eter, it  is  necessary  to  let  an  assistant  read  the  time,  both  on 
account  of  the  faintness  of  the  tick,  and  because  it  is  difficult 
to  carry  by  ear  a  beat  of  five  ticks  per  second. 

61.  The  image  of  the  Sun  seen  in  the  surface  of  the  mer- 
cury is  reversed  by  the  reflection  in  such  a  way  that  the 
apparent  upper  edge,  or  limb*  of  the  image  is  really  the 
image  of  the  lower  limb  of  the  Sun.  The  image  of  the  Sun 
received  by  way  of  the  index-glass  and  horizon-glass  is 
reversed  at  each  of  the  two  reflecting  surfaces,  and  is  finally 

*  The  word  limb  is  here  used  in  the  technical  sense,  in  which  it  means 
the  edge  of  the  visible  disk — of  the  Sun,  Moon,  or  other  heavenly  body. 


7O  GEODETIC  ASTRONOMY.  §  62. 

seen  as  if  no  reversal  had  taken  place.  If,  then,  one  makes 
the  lower  limb  of  the  index-glass  image  tangent  to  the  upper 
limb  of  the  mercury  image,  there  are  really  at  the  point  of 
tangency  two  coincident  images  of  the  same  point  of  the  Sun, 
namely,  the  lower  limb.  If  an  inverting  telescope  is  used 
both  images  are  reversed  again  in  addition  to  the  reversals 
stated  above.  The  record  of  observations  must  be  made  to 
show  which  limb  of  the  Sun  is  used  in  each  case.  The  object 
of  making  each  complete  set  of  observations  include  pointings 
upon  each  of  the  two  limbs  is  the  elimination  of  certain  errors, 
which  will  be  commented  upon  later  (§  74). 

62.  To  determine  the  index  error,  point  the  telescope  at 
the  Sun,  with  the  vernier  set  near  zero,  and  with  the  sextant 
in  such  a  position  that  a  line  in  the  plane  of  the  sextant  per- 
pendicular to  the  telescope  is  horizontal.  Make  the  direct 
and  reflected  images  of  the  Sun  tangent  to  each  other,  with 
the  zero  of  the  vernier  on  the  positive  part  of  the  graduated 
arc,  and  read  the  vernier.  Make  the  two  images  tangent  to 
each  other  in  the  reverse  position,  with  the  zero  of  the  vernier 
on  the  negative  portion  of  the  graduated  arc,  and  read  the 
vernier  again.  Repeat  the  process  two  or  three  times  for 
greater  accuracy.  Any  reading  on  the  positive  portion  of  the 
arc  evidently  gives  a  measure  of  the  Sun's  apparent  diameter. 
So  also  does  any  reading  on  the  negative  arc.  But  these  two 
measures  are  affected  equally  and  in  opposite  directions  by 
the  index  error.  Hence  both  the  index  error  and  the  Sun's 
diameter  become  known. 

The  Sun's  horizontal  diameter  is  measured  in  order  that 
the  results  may  not  be  affected  by  the  refraction  *  in  the  ver- 
tical plane.  In  making  readings  on  the  negative  arc  care 
must  be  taken  to  mentally  reverse  the  numbering  of  the 
graduations  on  the  vernier. 

*.  See  §§  67-69. 


§04.  COMPUTATION.  7 1 

EXAMPLE    OF    RECORD. 

63.  Determination  of  Time  by  Sextant  Observations  upon 
the  Sun. 

Astronomical  Station  No.  10. 

Longitude  2hi2m  west  of  Washington. 

Observer— J.  F.  H. 

Sextant — Greenough  No.  2083. 

Barometer  Reading,  25.56  in.  (Aneroid). 


Latitude  31°  19'  35" 

Date — October  14,  1892,  A.M. 

Chronometer — Dent  No.  2186. 


Thermometer  Reading,  17°  C. 


leading  of  Arc. 
On  Sun's 
Upper  Limb. 

Chronometer  Time. 

Reading  of  Arc. 
On  Sun's 
Lower  Limb. 

Chronometer  Time. 

64°  40'   oo" 

Ilh   Olm  35».  0 

65°  40'    oo" 

nu  07™  37s.  5 

65     oo     oo 

02      32  .0 

66     oo    oo 

08     36  .0 

65     20    oo 

03     31  .0 

66     20    oo 

09     35  .0 

65     40    oo 

04      29  .5 

66     40    oo 

10     33  -5 

66    oo    oo 

05      28  .5 

67    oo    oo 

ii     33  -o 

66     20    oo 

06      26  .O 

67      20      00 

12     33-5 

DETERMINATION  OF  INDEX  ERROR. 

By  Measurement  of  Sun's  Horizontal  Diameter. 
Readings  on  Arc.  Readings  on  Negative  Arc. 

32'    40"  31'    40" 

32      30  31      40 


Each  half  of  the  above  set  of  observations  was  computed 
separately.  The  computation  for  the  first  half  of  the  set,  on 
Sun's  upper  limb,  is  given  below.  The  explanation  of  the 
computation  follows  (§§  65-70). 

64.  Computation. 


Mean  reading  of  arc, 
Index  error  =  / 
Eccentricity   (not 
determined) 


65°  30'  oo" 

-28 

oo 


2AU                                             =  65     29    32 

Approx.  altitude  =  Au  =  32    44  46 

Sun's  semi-diameter      =  — 16  05 

Parallax  =  /  +07 

Refraction  =  /?              =  —   i    16 


Index  Error. 

Mean  reading  on  arc       =     32'  35" 
Mean  reading  on   neg- 
ative arc  =     31'  40" 


Diff.  =     oo'  55" 

4  Diff.  =  Index  error  =  /=— oo'  28" 


72  GEODETIC  ASTRONOMY.                               §65. 

Altitude  =  A  —  32    27  32 

Zenith  distance  =  C  =57    32  28 

Latitude  =  0  =  31    19  35                               log  cos  0=9.9315695 

Declination  =  d  =  —  S    29  33                                 log  cos  6=9.9952117 


49  08  9.9267812 

+  (0  -  5)]  =  48    40  48          log  sin  i[5  +  (0  -  (5)]=9.875659° 

-(0-<5)]  =     85140          log  sin  i[C-(0-(S)]=9.  1876329 

9.0632925 


log  sin8  it  =9.1365113 

it  =  21°  43'  06"        log  sin  it  =9.5682556 

t       =hour-angle  =  43    26   12 

TA  =apparent  solar  time  =  9*  o6m  I58.2 
E  =Equation  of  time  =  —  14  08.6 
TM  —  Mean  solar  time  =  8  52  06.6 
Tc  Time  by  chronom- 

eter 
=  Mean  of   six  given 

readings  =   n    04    oo  .3 

^/7V=Chronometer     cor- 

rection =  —  2    ii    53  .7 

Explanation  of  Record  and  Computation. 

65.  The  above  observations  were  made  at  uniform  inter- 
vals of  20'  on  the  sextant  arc  by  setting  the  vernier  before 
each  observation  to  that  exact  reading,  instead  of  taking  the 
readings  on  the  arc  after  each  random  pointing.  A  rough 
method  of  detecting  any  single  wild  observation  was  furnished 
by  the  fact  that  the  time  intervals  between  successive  read- 
ings must  be  nearly  the  same  throughout. 

The  mean  of  the  arc  readings  is  assumed  to  correspond  to 
the  mean  of  the  observed  chronometer  times.  This  would 
be  strictly  true  if  the  rate  of  change  of  the  Sun's  altitude,  as 
affected  by  refraction,  were  constant  during  the  period 
covered  by  each  half  set  of  observations.  The  rate  of  change 
varies  so  little  during  this  short  interval  that  the  error  intro- 
duced is  negligible. 


§  66.  PARALLAX   OF  SUN.  73 

The  method  of  computing  the  index  error  has  already 
been  indicated  (§  62). 

The  eccentricity  of  this  sextant  had  not  been  determined, 
but  was  known  to  be  small.  For  the  method  of  determining 
eccentricity,  see  §  76. 

Half  the  corrected  reading  of  the  sextant  is  the  approxi- 
mate altitude  of  the  Sun,  to  which  must  be  applied  the  cor- 
rections for  the  Sun's  semi-diameter,  for  parallax,  and  for 
refraction,  as  indicated  in  the  following  sections. 

The  pointings  were  made  upon  the  Sun's  upper  limb. 
But  the  position  of  the  Sun  as  given  in  the  Ephemeris  is  for 
the  Sun's  center.  Hence  the  angle  subtended  at  the  observer 
by  the  Sun's  semi-diameter  must  be  subtracted  to  reduce  the 
altitude  to  the  value  which  would  have  been  obtained  had 
the  observations  been  made  upon  the  center.  This  angular 
semi-diameter  of  the  Sun  is  given  in  the  Ephemeris  (pp. 
377-384  of  the  volume  for  1892)  for  every  day  at  Washington 
apparent  noon.  It  can  be  obtained  for  any  other  time  with 
all  needful  accuracy  by  an  interpolation  along  a  chord. 

Parallax. 

66.  Moreover,  since  the  right  ascension  and  declination  of 
the  Sun  define  its  position  on  the  celestial  sphere  as  seen 
from  the  Earth's  center,  it  is  necessary  to  reduce  the  observed 
altitude  to  what  it  would  have  been  had  the  observer  placed 
himself  at  the  center  of  the  Earth  and  had  used  the  same 
horizon  as  before. 

In  Fig.  7,  let  5  represent  the  position  of  the  Sun's  center, 
and  let  the  circle  BFG  represent  a  section  of  the  Earth  made 
by  a  plane  passing  through  the  observer,  at  B,  and  the  centers 
of  the  Sun  and  Earth  at  5  and  C,  respectively.  Let  BD 
represent  the  plane  of  the  observer's  horizon.  Let  CE  be 
parallel  to  BD.  Then  DBS  is  the  altitude  of  the  Sun  as  seen 


74  GEODETIC  ASTRONOMY.  §  66. 

by  an  observer  at  B.  ECS  is  the  altitude  that  would  be 
observed  by  him  if  he  were  at  the  Earth's  center  and  used  a 
horizon  plane  parallel  to  the  one  he  used  at  B.  The  differ- 
ence between  these  two  angles,  which  is  evidently  equal  to 
the  angle  BSC,  is  the  reduction  required. 

In  general,  the  parallax  of  an  object  is  its  apparent  dis- 
placement due  to  a  change  in  the  position  of  the  observer. 
The  parallax  of  any  celestial  object  is  the  difference  of  direc- 
tion of  two  straight  lines  drawn  to  it  from  two  different  points 
of  view.  It  is,  then,  the  angle  at  the  object  between  the  two 
straight  lines  drawn  to  it  from  the  two  points  from  which  it 
is  supposed  to  be  viewed.  The  word  parallax  y  unmodified, 
will  be  used  in  this  book  to  indicate  the  difference  of  direction 
of  a  celestial  object  as  seen  from  the  center  of  the  Earth  and 
from  a  station  on  the  surface.  The  horizontal  parallax  is  the 
parallax  for  an  object  which  is  in  the  horizon  of  the  observer. 
The  equatorial  horizontal  parallax  is  the  parallax  of  a  celestial 
object  seen  in  the  horizon  by  an  observer  at  a  station  on  the 
Earth 's  equator. 

In  Fig.  7,  if  S'  represents  a  position  of  the  Sun  in  the 
horizon  of  the  observer  at  B,  the  angle  BS'C  is  the  horizontal 
parallax  of  the  Sun.  It  is  the  angle  subtended  at  the  Sun  by 
the  radius  BC  of  the  Earth.  If  ph  is  the  horizontal  parallax 
of  the  Sun  in  seconds  of  arc,  r  is  the  radius  of  the  Earth,  and 
ds  is  the  distance  between  the  centers  of  the  Earth  and  Sun, 
then 

T» 

ph  =  -=—. =  about  9" (12) 

*/sm  \" 

The  exact  value  of  the  equatorial  horizontal  parallax  of 
the  Sun  is  given  in  the  Ephemeris  at  intervals  of  ten  days 
(p.  278  of  the  volume  for  1892).  The  different  radii  of  the 
Earth  are  so  nearly  equal  that  the  Sun's  horizontal  parallax 


§  6/.  REFRA  CTION.  7 5 

for  a  station  anywhere  on  the  surface  will  not  differ  from  that 
for  an  equatorial  station  by  more  than  o".O3, — a  quantity 
which  may  be  disregarded  for  our  present  purpose. 

Returning  to  the  figure,  let  BH  be  drawn  perpendicular 
to  CS.  If  p  is  the  parallax  of  the  Sun  at  any  position,  5, 
above  the  horizon,  then,  keeping  in  mind  that  /  and  ph  are 
very  small  angles, 


p-.ph  =  BH:  BC, 
or 

p=phCQsA (13) 

The  table  in  §  293,  abridged  from  Connaissances  des 
Temps,  serves  to  give  the  parallax  of  the  Sun  for  any  date 
and  altitude.  The  distance  of  the  Sun  is  so  nearly  the  same 
for  the  same  date  in  different  years  that  the  table  may  be  used 
for  any  year  for  several  centuries  to  come. 

Refraction. 

67.  The  path  of  a  ray  of  light  from  any  celestial  object  to 
an  observer  upon  the  Earth's  surface  is,  to  the  best  of  our 
knowledge,  a  straight  line  until  the  ray  enters  the  Earth's 
atmosphere.  From  that  point  onward  the  ray  encounters  at 
each  successive  element  of  its  path  a  stratum  of  air  which  is 
more  dense  than  the  stratum  left  behind,  since  the  density  of 
the  air  continually  increases  from  the  top  downward  with  the 
increasing  pressure  due  to  weight  of  the  superincumbent 
strata.  The  ray  is,  therefore,  continually  being  refracted,  or 
bent,  out  of  the  straight  line,  and  this  portion  of  its  path  is 
a  curve. 

The  two  general  laws  of  refraction  are:  That  when  a  ray 
passes  from  a  rarer  to  a  denser  medium  it  is  refracted  toward 
the  normal  to  the  separating  surface  by  an  amount  which  is  a 


76  GEODETIC  ASTRONOMY.  §  67. 

function  of  the  angle  between  the  ray  and  the  normal,  and  of 
the  densities  of  the  two  media;  and  that  a  plane  containing 
the  normal  and  the  original  ray  also  contains  the  refracted 
ray.  The  refraction  is  reversed  in  passing  from  a  denser  to 
a  rarer  medium. 

Let  Fig.  8  represent  a  portion  of  a  section  of  the  Earth 
and  its  atmosphere  made  by  a  plane  passing  through  the 
center  of  the  Earth  and  the  straight  portion  of  a  ray  of  light 
from  the  celestial  object  O  to  the  point  of  incidence,  a,  of  the 
ray  with  the  Earth's  atmosphere.  At  a  the  ray  is  refracted 
out  of  the  straight  line  Oa  to  a  new  direction  ab,  nearer  to 
the  normal  aC,  and  still  in  the  plane  OaC.  At  the  point  b  at 
which  the  ray  passes  to  a  denser  stratum  the  ray  is  again 
bent,  toward  the  normal  bC,  to  the  new  direction  be.  The 
ray  is  thus  refracted  at  the  successive  points  a,  b,  c,  d,  etc., 
remaining  always  in  the  plane  OaC  until  it  finally  reaches  the 
observer  at  A.  In  reality  the  path  is  a  continuous  curve, 
since  the  increase  of  density  is  continuous.  An  observer  at 
A  sees  the  object  in  the  direction  AO'  along  the  tangent  at 
A  to  the  path  of  the  ray.  The  angle  between  the  original 
direction  of  the  ray,  Oa,  and  its  final  direction,  O'A,  is  called 
the  astronomical  refraction,  or,  for  convenience,  simply  the 
refraction.  It  should  be  noted  that  the  refraction  as  de- 
scribed above  affects  altitudes  directly,  always  making  the 
observed  altitude  too  great,  but  has  no  effect  on  azimuth. 

In  the  above  treatment  it  is  assumed  that  the  various 
strata  of  air  are  horizontal  at  every  point.  For  a  statement 
of  the  extent  to  which  the  azimuth  is  affected  by  refraction 
because  of  the  error  of  the  above  assumption,  see  §  219. 

Even  if  the  law  of  variation  of  density  of  the  air  with  the 
height  were  a  simple  one,  the  computation  of  the  refraction 
would  be  a  complicated  process.  But  the  laws  governing  the 
variation  of  density  are  themselves  complicated,  and  not 


§68.  REFRACTION.  77 

thoroughly  known.  Hence  the  theory  of  refraction  is  long 
and  difficult.  The  treatment  of  refraction  in  this  book  will 
therefore  be  limited  to  an  explanation  of  the  tables  given  in 
§§  294-297,  which  contain  the  results  of  the  astronomer's 
investigations  in  convenient  form  for  the  engineer. 

68.  §  294  gives  the  mean  refraction*  or  refraction  under 
the  mean  conditions,  at  the  station  of  observation,  stated  at 
the  head  of  the  table,  viz.,  pressure  760  mm.  (=  29.9  in.) 
and  temperature  10°  C.  (=  50°  F.).  The  mean  refraction  is 
a  function  of  the  altitude,  since  the  refraction  of  a  ray  of  light 
in  passing  from  one  medium  into  another  is  a  function  of  the 
angle  between  the  ray  and  the  normal  to  the  dividing  surface. 

§  295  gives  the  factor,  CB,  by  which  the  mean  refraction 
must  be  multiplied  if  the  reading  of  the  barometer  is  not 
exactly  760  mm.  The  argument  of  this  table  is  the  barom- 
eter reading  uncorrected  for  temperature,  but  corrected  if 
necessary  for  its  index  error  when  its  temperature  is  10°  C. 
If  a  mercurial  barometer  is  used  with  a  brass  reading-scale 
it  is  necessary  to  apply  a  correction  to  the  reading  to  take 
account  of  the  difference  of  expansion  of  the  brass  scale  and 
the  mercury.  This  correction  is  usually  applied  directly  as  a 
correction  to  the  barometer  reading.  But  for  convenience,  in 
dealing  with  refractions,  it  has  been  expressed  as  a  correction 
to  the  mean  refraction,  and  is  given  in  §  297  in  terms  of 
the  reading  of  the  thermometer  which  is  attached,  to  the 

*This  table  was  obtained  by  combining  the  table  of  mean  refractions 
given  in  Doolittle's  Practical  Astronomy,  p.  628,  with  that  given  in  the 
Connaissances  des  Temps  for  1897,  p.  658.  The  values  of  Prof.  Doolittle's 
table  were  first  reduced  to  the  same  basis  as  those  of  the  French  table,  and 
then  the  indiscriminate  mean  of  corresponding  values  taken.  Prof.  Doo- 
little's table  is  said  to  be  based  upon  Bessel's  tables,  which  in  turn  were 
based  upon  certain  long  series  of  observations  as  reduced  by  Bessel,  using 
the  theory  of  refraction  elaborated  by  him.  The  French  tables,  on  the 
other  hand,  depend  upon  other  observations  and  upon  a  different  theory — 
that  of  Laplace. 


78  GEODETIC  ASTRONOMY.  §  69. 

barometer.  In  case  the  barometer  used  is  not  of  the  kind 
referred  to  above, — if,  for  example,  it  is  a  mercurial  barome- 
ter with  an  ivory  or  steel  scale,  or  if  it  is  an  aneroid, — the 
proper  corrections  (including  the  temperature  correction)  must 
be  applied  to  its  readings  to  reduce  to  millimeters  of  mercury 
at  10°  C.,  and  then  the  table  of  §  295  must  be  used,  but  that 
of  §  297  ignored. 

§  296  shows  the  factor,  C  ,  by  which  the  mean  refraction 
must  be  multiplied  to  take  into  account  the  temperature  of 
the  open  air  at  the  station  of  observation. 

To  sum  up,  the  refraction  R,  as  computed  from  these 
tables,  is 

R  =  RM(CB)(CD)(CA); (14) 

in  which  RM  is  the  mean  refraction  as  given  in  §  294,  and  CB, 
CD,  and  CA  are  the  factors  given  in  §§  295-297. 

The  density  of  the  air  along  the  line  of  sight,  and  there- 
fore the  refraction,  is  dependent  upon  the  pressure  and 
temperature  at  all  points  along  that  line.*  But  observations 
of  temperature  and  pressure  can  be  made  at  the  station  of 
observation  only.  The  refraction  is  expressed,  as  above,  in 
terms  of  the  pressure  and  temperature  at  the  station,  and  the 
temperature  and  pressure  are  assumed  to  vary  with  the  height 
according  to  certain  laws  which  depend  to  a  considerable 
extent  upon  theory  only. 

69.  It  is  in  order  here  to  inquire  what  errors  may  be 
expected  in  the  refractions  as  thus  computed.  The  values 
for  RM  as  derived  from  a  long  series  of  observations,  extending 
over  several  years,  at  one  observatory  when  compared  with 
the  corresponding  values  derived  from  a  similar  series  at 

*  It  is  also  dependent  to  a  very  small  extent  upon  the  humidity  of  the 
air, — to  so  small  an  extent,  however,  that  no  attempt  is  ordinarily  made  to 
take  the  humidity  into  account. 


§  69.  REFRACTION.  79 

another  observatory  are  found  to  differ  in  an  extreme  case  by 
as  much  as  0.5$,*  and  differences  of  half  that  amount  are  not 
infrequent.  No  table  of  refractions  can,  therefore,  be  de- 
pended upon  to  give  even  the  average  refraction  for  a  term 
of  years  at  an  arbitrarily  chosen  station  within  ¥^  part  of  its 
true  value.  The  error  of  the  refraction  for  any  one  observa- 
tion, as  derived  from  any  table,  must  be  uncertain  to  a  much 
greater  extent.  It  is  probable  that  the  refraction  corre- 
sponding to  any  single  observation  as  computed  by  the  use  of 
any  available  tables  or  formulae  will  often  be  in  error  by  more 
than  i$.f  This  amounts  to  o^.o  to  0^.5  for  altitudes  from 
90°  to  50°,  o".5  to  i".5  for  altitudes  from  50°  to  20°,  i" '.5  to 
3//.o  for  altitudes  from  20°  to  10°,  and  increases  rapidly  for 
smaller  altitudes.  From  considerations  which  may  not  be 
entered  into  here  it  seems  probable  that  the  refractions  above 
50°  of  altitude  are  subject  to  greater  uncertainty  than  that 
indicated  above. 

Theories  of  astronomical  refraction  all  depend  upon  the 
assumption  that  surfaces  of  equal  density  in  the  atmosphere 
are  everywhere  horizontal,  and  that  the  density  varies  with  the 
height  according  to  some  fixed  law.  If  one  reflects  upon  the 
unceasing  changes  of  pressure  in  the  air  as  indicated  by  the 
winds  and  by  the  fluctuating  barometer,  upon  the  large  and 
irregular  changes  in  temperature  near  the  Earth's  surface, 
and  upon  the  continual  changes  in  humidity  indicated  by  the 

*  Astronomical  Papers,  American  Ephemeris  and  Nautical  Almanac, 
vol.  n.  Part  vi. 

f  This  is  the  reason  why  the  more  cumbersome  and  more  accurate 
method  of  computing  the  refraction  by  Bessel's  factors  has  been  omitted 
in  this  book.  With  the  limited  number  of  observations  which  the  engineer 
usually  makes  in  determining  any  one  quantity,  the  actual  accuracy  of  the 
final  result  attained  by  the  use  of  the  tables  given  will  not  differ  sensibly 
from  that  which  would  be  obtained  with  a  greater  expenditure  of  time  from 
other  tables  or  formulae. 


8O  GEODETIC  ASTRONOMY.  §  /O. 

evanescent  clouds,  the  large  margin  of  uncertainty  in  the 
refraction  stated  above  seems  not  only  reasonable,  but  in- 
evitable. If,  however,  more  direct  evidence  is  needed  to 
convince  one,  it  will  usually  be  found  in  making  any  long 
series  of  astronomical  observations  in  our  climate.  For,  on 
some  nights,  in  attempting  to  make  an  accurate  pointing  upon 
a  star  with  a  telescope  it  may  be  observed  to  be  apparently 
oscillating  irregularly  through  a  range  of  three  or  four  seconds 
(say)  about  its  mean  position  on  account  of  momentary 
changes  in  refraction. 

Derivation  of  Formula. 

70.  The  three  corrections  for  Sun's  semi-diameter,  paral- 
lax, and  refraction  being  applied  to  the  approximate  altitude 
Au,  the  result  is  the  measured  altitude,  A,  of  the  Sun's 
center.  The  complement  of  this  altitude,  or  the  zenith  dis- 
tance, £,  of  the  Sun's  center,  is  a  side  of  the  spherical  triangle 
(Fig.  9)  Sun-zenith-pole,  upon  the  celestial  sphere.  The  arc 
zenith  to  pole  of  that  triangle  is  the  complement  of  the  lati- 
tude, which  is  supposed  to  be  known.  The  declination  of 
the  Sun  at  the  instant  of  observation  may  be  obtained  from 
the  Ephemeris  by  interpolation  along  a  tangent  as  indicated 
in  §  35-  The  necessary  interpolation  extends  over  the  inter- 
val from  the  nearest  Washington  mean  noon.  To  obtain  this 
interval  one  may  assume  an  error  for  the  chronometer  (to  be 
checked  later).  For  this  purpose  it  is  only  necessary  to 
know  the  error  within  one  minute.  For  example,  in  the 
computation  in  hand  it  was  known  from  previous  observa- 
tions that  the  error  of  the  chronometer  on  Washington  mean 
time  was  about  om.  The  Washington  mean  time  of  observa- 
tion was  then  nh  O4m,  and  the  interpolation  interval  56™. 
The  complement  of  the  declination  is  the  third  side,  Sun  to 
pole,  of  the  spherical  triangle  Sun-zenith-pole.  The  three 


§  70.  DERIVATION   OF  FORMULA.  8 1 

arcs  of  this  spherical  triangle  being  known,  the  angle  at  the 
pole,  which  is  the  hour-angle  of  the  Sun,  may  be  computed 
by  spherical  trigonometry. 

Let  At,  Bt1  and  Ct  be  the  angles  of  any  spherical  triangle, 
and  at,  bt,  and  ct  the  sides  opposite,  respectively.  From 
spherical  trigonometry 


sn       t  = 


sin      sm 


u-  u 
in  which 


In  the  triangle  Sun-zenith-pole  let  At  be  the  angle  at  the 
pole,  namely,  the  hour-angle  of  the  Sun,  t.  Let  bt  be  the 
side  zenith  to  pole  (=  90°  —  0),  and  ct  be  the  side  Sun  to 
pole  (=  90°  —  #).  at  must  be  £,  the  zenith  distance  of  the 
Sun.  Making  the  substitutions  indicated,  squaring  both 
members  of  the  equation,  and  simplifying,  there  is  obtained 


. 

cos  0  cos  d 

by  the  use  of  which  the  hour-angle  may  be  computed  as  in- 
dicated in  §  64. 

The  hour-angle  subtracted  from  I2h,  the  Sun  being  east 
of  the  meridian,  is  the  local  apparent  solar  time,  TA.  The 
equation  of  time,  E  (see  §  20),  may  be  obtained  from  the 
Ephemeris  with  sufficient  accuracy  by  an  interpolation  along 
a  chord  from  the  nearest  Washington  apparent  noon.  TA  +  E 
is  the  local  mean  solar  time,  TM.  ATCt  the  correction  to  the 
chronometer  to  give  local  mean  time,  is  evidently  the  differ- 
ence between  Tc,  the  reading  of  the  chronometer,  and  TM. 
It  should  be  kept  definitely  in  mind  that  ATC  is  strictly  the 


82  GEODETIC  ASTRONOMY.  §  ?2. 

correction  to  the  reading  of  the  chronometer  only  at  the  one 
instant  when  the  reading  of  the  chronometer  is  Tc.  At  any 
other  instant  the  chronometer  will  have  a  different  correction, 
depending  upon  its  rate. 

If  the  chronometer  used  in  the  sextant  observations  keeps 
mean  time,  and  it  is  proposed  to  obtain  from  it  the  error  of  a 
sidereal  chronometer,  the  two  chronometers  should  be  com- 
pared by  the  method  indicated  in  §  250. 

Discussion  of  Errors. 

71.  The  various  errors  which  affect  the  final  result  in  any 
astronomical  observation  may  be  grouped  in  three  classes: 
1st,  external  errors,  or  errors  arising  from  conditions  outside 
the    instrument    and   observer;    2d,    instrumental  errors,    or 
errors   due  to   the   instrument,   arising  from  lack   of  perfect 
adjustment,  from  imperfect  construction,  from  instability  of 
the  relative  positions  of  different  parts,  etc. ;  3d,   observer  s 
errors,    or    errors    due    directly   to    the    inaccuracies   of    the 
observer,  arising  from  his  unavoidable  errors  in  judgment  as 
to  what  he  sees  and  hears,  and  from  the  fact  that  his  nerves 
and  brain  do  not  act  instantaneously.      By  the  phrase  errors 
of  observation  is  meant  the  errors  arising  from  all  these  sources 
combined. 

External  Errors. 

72.  Following  the  order  indicated  above,  let  us  first  con- 
sider the  errors  arising  from  conditions  outside  the  instrument 
and  observer. 

The  accuracy  of  a  determination  of  time  from  observations 
upon  the  Sun  depends  largely  upon  the  part  of  the  day  at 
which  the  observations  are  made.  Near  apparent  noon  the 
altitude  of  the  Sun  is  changing  quite  slowly.  A  few  hours 
later  or  earlier  the  change  of  altitude  is  comparatively  rapid. 


§  72.  EKKOKS.  83 

Evidently,  the  effect  of  a  given  error  in  the  measured  altitude 
upon  the  computed  time  will  be  less  the  greater  is  the  rate  of 
change  of  the  altitude.  It  may  be  shown  from  the  differen- 
tial formulae  *  applicable  to  the  spherical  triangle  used  in  the 
preceding  computation  (§  70),  that  this  rate  of  change  is 
greatest  when  the  Sun  is  in  the  prime  vertical,  or  when  it  is 
nearest  the  prime  vertical  in  case  it  does  not  reach  it  while 
above  the  horizon.  This  condition  by  itself  would  fix  the 
most  favorable  time  for  observations  at  sunrise  or  sunset 
during  the  months  when  the  Sun  is  south  of  the  equator,  and 
from  three  to  six  hours  from  the  meridian  during  the 
remainder  of  the  year,  for  nearly  all  points  in  the  United 
States. 

Another  condition,  however,  must  also  be  considered  in 
determining  the  most  favorable  time  for  observing.  As  indi- 
cated in  §  69,  the  uncertainty  in  the  computed  refraction 
increases  rapidly  as  the  altitude  diminishes.  In  so  far,  then, 
as  the  refraction  is  concerned,  the  nearer  to  apparent  noon  the 
observations  are  made  the  better.  Taking  both  conditions 
into  account  the  most  favorable  time  for  observing  is  from 
two  to  four  hours  from  the  meridian.  Within  these  limits 
and  for  stations  in  the  United  States  (excluding  Alaska)  the 
rate  of  change  of  altitude  is  from  5  to  14  seconds  of  arc  per 
second  of  time,  and  the  error  in  the  derived  chronometer  cor- 
rection arising  from  the  uncertainty  of  the  computed  refrac- 
tion, adopting  the  estimate  of  that  uncertainty  as  given  in 
§  69,  may  be  from  os.O2  under  the  most  favorable  conditions 
(latitude  24°  30',  midsummer)  to  os.5  for  an  observation  at 
two  hours  from  the  meridian  in  latitude  49°  in  midwinter,  or 
even  2s  if  this  last  observation  is  made  near  sunrise  or  sunset. 


*See  Chauvenet's  Astronomy,  vol.  i.  pp.  213,214;  or  Doolittle's    Prac- 
tical Astronomy,  p.  223. 


84  GEODETIC  ASTRONOMY.  §  73- 

The  position  of  the  Sun  is  so  well  determined  that  there 
is  no  sensible  error  in  the  result  from  that  cause. 


Instrumental  Errors. 

73.  If  the  telescope  is  not  perfectly  focused  upon  the 
Sun,  or  if  the  colored  glasses  introduced  make  the  images  of 
the  Sun  very  dim,  or  leave  them  too  bright  to  be  gazed  at 
with  comfort,  there  is  a  tendency  to  see  the  images  either 
larger  or  smaller  than  they  really  are,  and  so  to  misjudge  the 
position  of  tangency.  This  is  not  eliminated  by  the  determi- 
nation of  index  error,  as  described  in  §  62,  for  the  effect 
would  be  to  increase  (or  decrease)  both  plus  and  minus  read- 
ings by  the  same  amount,  and  so  leave  the  computed  index 
error  unchanged.  But  it  is  eliminated  by  taking  half  of  the 
observations  on  the  upper  limb  of  the  Sun  and  half  on  the 
lower  limb,  as  shown  in  the  set  of  observations  given  in  §  63. 

The  inclination  of  the  index-glass  to  the  perpendicular  to 
the  sextant  plane,  and  the  inclination  of  the  axis  of  the  tele- 
scope to  that  plane,  combine  to  produce  an  error  which  varies 
as  the  tangent  of  one-quarter  of  the  measured  angle.*  The 
error  of  adjustment  of  the  index-glass  and  of  the  telescope 
may  each  be  made  less  than  5'  by  the  methods  given  in 
§§  54>  S^.  If  each  is  5',  the  maximum  error  introduced  into 
a  measured  angle  of  120°  is  4/x.o,  and  for  other  angles  in  the 
ratio  indicated  above.  This  error  is  therefore  small  provided 
the  adjustments  are  carefully  made  and  frequently  verified, 
but  it  is  sensibly  a  constant  affecting  the  mean  of  a  set  of 
observations  made  at  nearly  the  same  reading  of  the  arc.  If 
the  telescope  is  parallel  to  the  plane  of  the  sextant,  but,  in 
observing,  the  contacts  are  made  with  the  images  out  of  the 
center  of  the  field,  the  sight  line  is  inclined  to  the  plane  of  the 

*  Chauvenet's  Astronomy,  vol.  n.  p.  116. 


§  73-  ZKXORS.  85 

sextant,  and  the  effect  on  the  measured  angle  is  the  same  as 
if  the  telescope  were  so  inclined.  It  is  important,  therefore, 
that  every  observation  should  be  made  nearly  in  the  middle 
of  the  field  of  the  telescope. 

If  the  horizon-glass  is  not  perpendicular  to  the  plane  of 
the  sextant,  the  error  introduced  is  greater  the  smaller  the 
angle  observed,  and  will  ordinarily  be  appreciable  only  in  the 
determination  of  the  index  error.  In  determining  the  index 
error  by  observing  the  Sun's  semi-diameter,  the  error  in  any 
one  reading  from  this  cause  will  be  less  than  \"  even  if  the 
horizon-glass  is  inclined  as  much  as  50"  to  its  normal  position 
(which  is  about  the  maximum  error  of  this  adjustment  made 
as  indicated  in  §  55).  This  error  is  eliminated  from  the 
derived  index  correction,  for  both  positive  and  negative  read- 
ings are  numerically  too  small  by  the  same  amount. 

If  the  center  about  which  the  index-arm  swings  does  not 
coincide  with  the  center  about  which  the  graduated  arc  is 
described,  an  error  due  to  this  eccentricity  will  be  introduced 
into  every  reading.  The  magnitude  of  this  error  will  evi- 
dently depend  upon  the  size  of  the  angle  measured  as  well  as 
upon  other  conditions.  See  §  76  for  the  method  of  deter- 
mining, and  correcting  for,  eccentricity.  . 

The  errors  treated  in  the  last  three  paragraphs  are  func- 
tions of  the  angle  measured,  but  are  constant  for  a  given 
reading  of  the  sextant  so  long  as  the  condition  of  the  instru- 
ment remains  unchanged.  Their  effect  may  therefore  be 
eliminated  almost  wholly  from  the  final  result  in  determining 
time  by  measured  altitudes  of  the  Sun,  by  making  observa- 
tions both  in  the  forenoon  and  afternoon  at  about  the  same 
altitude.  The  computed  altitude  will  be  too  great,  or  too 
small,  by  the  same  amount  in  both  cases,  if  the  two  altitudes 
are  equal,  and  one  computed  time  will  be  as  much  too  late  as 
the  other  is  too  early.  This  procedure  will  also  ^eliminate  the 


86  GEODETIC  ASTRONOMY.  §  74. 

error  in  the  computed  time  arising  from  an  error  in  the 
assumed  latitude. 

The  errors  arising  from  changes  in  the  relative  position  of 
different  parts  of  the  sextant  due  to  stresses  or  to  changes  of 
temperature  are  probably  small  in  comparison  with  the  other 
errors  considered  under  the  next  heading.  So  also  are  the 
errors  of  graduation  of  the  sextant  arc. 

The  error  which  may  arise  from  either  or  both  glasses  of 
the  horizon  roof  being  prismatic  instead  of  plane,  may  be 
eliminated  by  reversing  the  roof  when  half  the  observations 
have  been  taken. 

To  avoid  errors  arising  from  the  prismatic  form  of  the 
shades,  some  instrument-makers  provide  a  contrivance  by 
which  the  colored  shades  may  be  rotated  180°  from  their 
original  position;  but  it  is  better  to  use  colored  shades 
between  the  eyepiece  and  the  eye  instead  of  the  colored 
shades  in  front  of  the  index  and  horizon  glasses.  A  shade  in 
this  position  may  be  of  a  prismatic  form  without  vitiating  the 
observed  results. 

Observer's  Errors. 

74.  The  errors  which  are  classed  as  instrumental  depend 
to  a  considerable  extent  upon  the  care  and  judgment  with 
which  the  sextant  is  manipulated.  But  aside  from  the 
manipulation,  which  is  an  important  as  well  as  a  difficult 
portion  of  the  observer's  duty,  the  final  result  also  depends 
upon  his  estimates  of  the  positions  of  contact  of  the  two 
images  and  of  the  chronometer  times  of  those  contacts. 

His  estimate  of  the  position  of  contact  is  subject  to  both 
an  accidental  and  a  constant  error.*  The  accidental  error 

*  A  constant  error  is  one  which  has  the  same  effect  upon  all  the  observa- 
tions of  the  series,  or  portion  of  a  series,  under  consideration.  Accidental 
errors  are  not  constant  from  observation  to  observation;  they  are  as  apt  *o 


§  74-  £XAOAS.  87 

depends  mainly  upon  the  personality  and  experience  of  the 
observer  and  the  care  with  which  he  observes,  but  also  to  a 
certain  extent  upon  the  steadiness  of  the  refraction,  the  power 
of  the  telescope,  the  brightness  and  definition  of  the  images, 
and  the  physical  conditions  affecting  the  observer's  comfort. 
A  probable  error  of  ±  14"  seems  from  various  recorded  series 
of  observations  to  be  a  fair  estimate  of  the  accidental  error  in 
a  single  measurement  of  the  Sun's  double  altitude  by  an 
experienced  observer  with  an  ordinary  sextant  under  average 
conditions.  The  accidental  error  in  the  mean  of  twelve 
observations  constituting  a  set  is  on  this  basis  ±  14" 
-f-  4/i2  =  4".  This  corresponds,  for  observations  taken  in 
the  United  States  when  the  Sun  is  observed  from  two  to  four 
hours  from  the  meridian,  to  ±  o8. 15  to  ±  O8.4O.  The  con- 
stant error  of  the  observer's  estimate  of  the  position  of  contact 
is  eliminated  from  the  mean  for  a  set  if  half  the  observations 
are  taken  upon  the  Sun's  upper  limb  and  half  upon  the  lower. 
The  observer's  estimate  of  the  time  of  contact  is  also 
subject  to  both  an  accidental  and  a  constant  error.  The 
accidental  error,  judging  from  time  observations  made  with  a 
transit  instrument,  is  about  ±  o8. 1  for  a  single  observation, 
or  ±  08.03  for  a  mean  of  twelve  observations  constituting  a 
set, — a  small  error  as  compared  with  that  arising  from  the 
uncertainty  of  the  position  of  contact.  The  constant  error 
made  in  estimating  the  time,  or  personal  equation*  of  the 
observer,  may  be  as  great  as  O8.5  for  some  men.  It  affects 
all  the  observations  of  a  set  alike. 

be  minus  as  plus,  and  they  presumably  follow  the  law  of  error  which  is 
the  basis  of  the  theory  of  least  squares.  It  is  then  the  effect  of  accidental 
errors  upon  the  final  result,  which  may  be  diminished  by  continued  repe- 
tition of  the  observations  and  by  the  least  square  methods  of  computa- 
tion, whereas  the  effect  of  constant  errors  must  be  eliminated  by  other 
processes. 
*  See  §  125. 


88  GEODETIC  ASTRONOMY.  §  76. 

Error  of  the  Computed  Time. 

75.  The  mean  result  from  a  set  of  observations  such  as 
that  given  in  §  63  is  subject,  then,  to  an  accidental  error  of 
about  ±  O8.25,   on  an  average,  arising  almost  entirely  from 
the  observer's  accidental  errors.      It  is  also  subject  to  an  error 
which  is  constant  for  the  set,  arising  from  uneliminated  instru- 
mental errors  and  the  error  of  the  computed  refraction  (neglect- 
ing for  the  time  being  the  personal  equation  of  the  observer). 
A  fair  estimate  of  this  constant  error  under  average  field  con- 
ditions seems  to  be  ±  O8.25.     This  makes  the  probable  error 

of  the  result  from  the  set  about  V  0.25"  -f-  0.25'  =  ±  os.35, 
aside  from  personal  equation.  It  is  evident  from  the  above 
estimate  that  increasing  the  number  of  observations  in  a  set, 
or  number  of  sets  taken  under  the  same  circumstances, 
diminishes  the  final  error  but  little  (only  one  term  under  the 
radical  above  being  reduced).  The  constant  instrumental 
error  may,  however,  be  almost  entirely  eliminated  by  making 
observations  at  about  the  same  altitude  in  both  forenoon  and 
afternoon  as  indicated  in  §  73.  There  is  no  feasible  way  of 
eliminating  the  personal  equation  error  in  the  field. 

The  above  estimate  of  the  errors  from  various  sources 
is  believed  to  be  a  fair  one  for  average  conditions.  A 
special  investigation  for  a  particular  observer  and  set  of  con- 
ditions may  show  errors  either  somewhat  smaller  or  much 
larger  than  those  indicated. 

Correction  for  Eccentricity. 

76.  Unless  the  center  about  which  the  index-arm  swings 
coincides  exactly  with  the  center  of  the  graduation,   every 
sextant  reading  will  be  in  error  by  the  effect  of  this  eccen- 
tricity (as  noted  in  §  73),  which  effect  is  different  for  readings 
taken  on  different  parts  of  the  arc.     To  eliminate  the  effect 


76.  ECCENTRICITY.  89 


of  eccentricity  upon  the  sextant  readings  one  may  proceed  as 
follows  : 

First.  The  values  of  angles  as  measured  with  the  sextant 
may  be  compared  with  their  true  values  determined  in  some 
other  way. 

For  example,  the  angles  between  certain  terrestrial  objects 
may  be  measured  with  the  sextant  and  then  with  a  good 
theodolite.  In  making  this  comparison  it  must  be  kept  in 
mind  that  a  theodolite  as  ordinarily  used  measures  horizontal 
and  vertical  angles,  while  the  sextant  measures  directly  the 
angle  between  the  two  objects  in  the  plane  (horizontal, 
oblique,  or  vertical)  passing  through  the  two  objects  and  the 
sextant.  Also,  in  this  case,  the  sextant  parallax,  §  53,  must 
be  taken  into  consideration  unless  the  objects  are  very 
distant. 

The  angular  distance  between  two  known  stars  may  be 
observed  and  compared  with  its  value  as  computed  from  the 
known  right  ascensions  and  declinations  of  the  stars,  corrected 
for  the  effect  of  refraction  at  the  time  of  observation.  This 
computation  will  be  found,  unfortunately,  to  be  rather 
laborious. 

Or,  the  altitude  of  a  known  star  (or  of  the  Sun)  may  be 
measured  at  a  known  time  at  a  station  of  which  the  latitude 
is  known.  The  true  altitude  of  the  star  may  be  computed, 
and  becomes  comparable,  after  correction  for  refraction,  with 
that  measured  with  the  sextant. 

Second.  For  each  sextant  observation  which  is  compared 
with  a  known  angle  an  observation  equation  of  the  form 

Jx  +  Ky  +  v  =  DA    .....     (17) 

is  formed,  in  which  xy  y,  and  v  are  unknowns  to  be  deter- 
mined, and  DA  —  Bt  —  Qm  is  the  difference  between  the  true 


90  GEODETIC  ASTRONOMY.  §  7/. 

value  of  the  angle  6t  and  the  measured  value  0m.     6m  is  the 
reading  of  the  sextant  corrected  for  index  error. 

e        8  e 

J  =  sin  --  cos  -        and      K  =  sin2  — .    .      .     (18) 

Third.  The  most  probable  values  of  x,  y,  and  v  are 
determined  from  these  observation  equations  by  the  method 
of  least  squares. 

Fourth.  These  values  of  x,  y,  and  v  may  now  be  substi- 
tuted in  equation  (17)  and  a  table  of  corrections  computed  by 
substituting  o°,  10°,  20°,  .  .  .  successively  for  6m,  the  corre- 
sponding computed  values  of  DA  being  evidently  the  correc- 
tions for  eccentricity  which  must  be  applied  to  measured 
angles.* 

So  many  and  such  accurate  observations  are  required  for 
a  satisfactory  determination  of  the  eccentricity  of  a  sextant 
that  it  will  usually  be  found  more  convenient  to  eliminate  the 
effect  of  eccentricity  upon  time  observations  by  observing 
both  in  the  forenoon  and  afternoon  with  the  Sun  at  about  the 
same  altitude,  as  indicated  in  §  73.  But  in  sextant  observa- 
tions for  latitude  a  special  determination  of  the  eccentricity 
is  necessary  if  the  highest  attainable  degree  of  accuracy  is 
desired. 

Other  Uses  of  the  Sextant. 

77.  In  determining  time  with  a  sextant  by  the  preceding 
method  the  latitude  is  supposed  to  be  known.  If,  conversely, 
the  time  of  observation  of  the  altitude  of  the  Sun  (or  a  star) 

*  This  method  of  determining  the  corrections  to  be  applied  for  eccen- 
tricity, which  is  here  given  in  condensed  form,  and  without  the  derivation 
of  the  formulae,  will  be  found  treated  in  full  in  Doolittle's  Practical 
Astronomy,  pp.  196-206.  Certain  refinements  there  given,  which  add  much 
to  the  labor  of  computation  and  little  to  the  accuracy  of  the  computed 
result,  have  here  been  omitted. 


§  79-  MISCELLANEOUS.  9! 

is  known,  the  latitude  of  the  station  may  be  computed  by  the 
method  of  §  171,  or  by  that  of  §  172,  if  the  observation  is 
made  near  the  meridian.  The  most  favorable  time  for  thus 
determining  the  latitude  by  observations  upon  the  Sun  is 
about  apparent  noon,  for  then  the  altitude  is  changing  very 
slowly,  and  hence  an  error  in  time  will  have  but  little  influence 
on  the  computed  result.  The  refraction  is  also  a  minimum 
at  apparent  noon. 

78.  At  sea,   observations  with  the  sextant  for  time  are 
usually  made  in  the  middle  of  the  forenoon  and  of  the  after- 
noon, and  for  latitude  at  apparent  noon.     To  make  this  lati- 
tude observation  the  Sun  is  watched  with  a  sextant  for  a  few 
minutes  before  apparent  noon,  as  its  altitude  increases  slowly 
at  a  diminishing  rate.     The  observation  is  made  when   the 
altitude  stops  increasing  and  is  at  its  maximum.     With  suffi- 
cient accuracy  it  may  be  assumed  that  the  Sun  is  then  on  the 
meridian,  and  that  therefore  the  latitude  is  the  declination  of 
the  Sun  plus  its  south  zenith  distance.      In  observing  at  sea 
the  natural  horizon  is  used,  and  an  allowance  must  be  made 
for  the  dip  of  the  horizon,  or  downward  inclination  of  the  line 
of  sight  to  the  apparent  horizon,  due  to  the  height  of  the  sex- 
tant above  the  surface  of  the  sea  (see  table,  §  298  *).     The 
observation  of  latitude  and  of  local  time  serves  to  locate  the 
observer  at  sea,  provided  he  also  knows  the  Greenwich  time 
of  the  observation.  JjjThis  ^ast  he  dbtains  from  the  known  rate 
of  his  chronometer  and  its  known  error  on  Greenwich  time  at 
some  previous  date. 

79.  An   observation  at  sea  of  the  altitude,  at  a  known 
instant    of    Greenwich    time,    of    any   celestial   object    (Sun, 
Moon,  planet,  or  star)  serves  to  locate  the  observer  upon  an 

*This  table  is  reproduced,  with  a  slight  extension,  from  Chauvenet's 
Astronomy.     It  is  computed  for  a  mean  state  of  the  atmosphere. 


92  GEODETIC  ASTRONOMY.  §  80. 

arc  of  a  small  circle  on  the  Earth's  surface,  of  which  the  pole 
is  at  a  point  in  the  line  joining  the  object  and  the  Earth's 
center,  and  of  which  the  polar  distance  is  equal  to  the 
observed  zenith  distance  of  the  object.  This  small  circle,  or 
such  a  portion  of  it  as  is  necessary,  may  be  plotted  on  a 
sphere  or  chart.  A  second  such  observation  on  an  object  in 
some  other  azimuth  serves  to  locate  the  observer  on  another 
small  circle  intersecting  the  first  in  two  points.  These  two 
points  of  intersection  are  usually  so  far  apart  that  there  is  no 
difficulty  in  discriminating  between  them,  and  the  observer's 
position  becomes  definitely  known.  This  process  of  deter- 
mining a  position  at  sea  is  known  as  Sumner's  method.  For 
a  more  complete  statement  of  this  method  see  Chauvenet's 
Astronomy,  vol.  I.  pp.  424-428. 

80.  If,  for  the  purpose  of  determining  local  time,  an  obser- 
vation is  taken  upon  a  star  east  of  the  meridian,  and  the 
observation  is  repeated  west  of  the  meridian  at  the  same 
reading  of  the  sextant,  the  computation  of  time  may  be  made 
independently  of  any  knowledge  of  the  index  error,  eccen- 
tricity, or  other  errors  of  the  sextant,  and  independently  of 
any  computation  of  the  refraction,  upon  the  assumption  that 
these  quantities  retain  the  same  values  at  the  second  observa- 
tion which  they  had  at  the  first,  and  that  therefore  the  two 
observations  are  made  at  the  same  (unknown)  altitude.  Dur- 
ing such  an  interval  of  a  few  hours,  the  declination  of  any 
star  is  for  the  present  purpose  sensibly  constant.  Upon  these 
assumptions  it  may  be  shown  that  the  mean  of  the  two 
observed  chronometer  times  is  the  chronometer  time  corre- 
sponding to  the  transit  of  the  star  across  the  meridian.  (Let 
the  student  prove  this.)  If  the  object  observed  is  a  planet, 
the  Moon,  or  the  Sun,  the  same  method  of  computation  may 
be  used,  but  it  will  be  necessary  to  apply  a  correction  for  the 


§  8 1.  MISCELLANEOUS.  93 

change  of  declination  during  the  interval  between  the  two 
observations.* 

The  advantages  of  this  method  are  the  ease  and  simplicity 
of  the  computation.  Its  disadvantage  is  the  liability  of 
losing  the  second  observation  on  account  of  clouds  or  other 
hindrances.  If  observations  are  taken  within  one  hour  (say) 
of  the  same  hour-angle  east  and  west  of  the  meridian  respec- 
tively, are  computed  as  indicated  in  §  64,  and  the  mean 
taken,  the  elimination  of  errors  is  almost  as  complete,  advan- 
tage may  be  taken  of  temporary  breaks  in  the  clouds,  and  the 
observer  is  not  put  to  the  inconvenience  of  being  ready  at 
some  particular  moment.  The  computation  will  consume  a 
little  more  time. 

81.  The  Covarrubias  method  of  observing,  developed  by 
the  Mexican  astronomer  of  that  name,  serves  to  eliminate  the 
instrumental  errors,  and  accomplishes  that  purpose  without 
the  necessity  of  the  long  wait  between  observations  which  is 
required  in  the  method  stated  above.  Two  stars  are  selected 
which  are  several  hours  apart  in  right  ascension,  and  have 
declinations  not  very  different.  At  a  certain  time  each  night, 
which  is  first  estimated  roughly  by  the  observer,  these  two 
stars  will  for  an  instant  be  at  the  same  altitude,  one  east  and 
the  other  west  of  the  meridian.  A  few  minutes  before  this 
time  he  observes  one  of  the  stars,  noting  the  chronometer 
time  and  the  sextant  reading.  He  then  turns  to  the  second 
star,  which  he  finds  approaching  the  same  altitude,  and 
observes  the  chronometer  time  at  which  the  sextant  reading, 
and  therefore  the  altitude,  is  the  same  for  this  second  star  as 
that  before  observed  upon  the  first  star.  From  this  observa- 
tion of  the  two  chronometer  times  at  which  the  two  stars  reach 

*  For  the  computation  of  this  correction,  see  Doolittle's  Practical 
Astronomy,  pp.  230,  231;  Chauvenet's  Astronomy,  vol.  I.  pp.  198-201;  or 
Loomis'  Practical  Astronomy,  pp.  126-130- 


94  GEODETIC  ASTRONOMY.  §82. 

the  same  altitude  the  error  of  the  chronometer  may  be  com- 
puted independently  of  any  knowledge  of  the  exact  absolute 
value  of  that  altitude.  This  method  will  sometimes  be  found 
desirable,  especially  in  case  the  only  available  sextant  is  an 
inferior  one,  or  has  been  damaged  to  such  an  extent  that  its 
indications  are  unreliable.  It  is  not  developed  in  detail  here 
for  lack  of  space.  For  a  complete  statement  of  the  method 
see  "  Nuevos  Metodos  Astronomicos  para  determinas  la  hora, 
el  azimut,  la  latitude  y  la  longitude";  F.  D.  Covarrubias, 
Mexico,  1867. 

QUESTIONS  AND   EXAMPLES. 

82.  i.  Prove,  using  figures  if  necessary,  that  the  test  as 
given  in  §  54  for  determining  whether  the  index-glass  is  per- 
pendicular to  the  plane  of  the  sextant  is  valid. 

2.  Explain  why  two  images  of  the  same  object  cannot  be 
made  to  coincide  in  the  sextant  telescope  if  the  index-glass  is 
in  perfect  adjustment  but  the  horizon-glass  is  inclined  to  the 
plane  of  the  sextant  (see  §  55).      Explain  also  how  it  is  possi- 
ble that  such  coincidence  may  be  secured  if  both  the  index 
and  horizon  glasses  are  inclined.     Why  is  it  advisable  to  make 
certain  of  the  index-glass  adjustment   before  adjusting  the 
horizon-glass  ? 

3.  Suppose  that  the  index  correction  of  a  certain  sextant 
is  found  to   be  —  15" '.     Through  what   angle   and   in  what 
direction  must  the  horizon-glass  be  rotatecl  to  make  the  cor- 
rection zero  ? 

4.  Show  that  the  errors  due  to  the  inclination  to  the  plane 
of  the  sextant  of  the  sight  line  and  of  the  index  glass  are 
not  eliminated  by  the  process  of  eliminating  the  index  error 
indicated  in  §  62. 

5.  The  radius  of  the  graduated  circle  of  a  certain  sextant 
is  4  in.     What  linear  movement  of  the  vernier  corresponds 


§  82.  QUESTIONS  AND   EXAMPLES.  95 

to  a  change  of  10"  in  its  reading  ?     Explain  the  need  of  the 
caution  in  the  last  sentence  of  §  58. 

6.  Explain,    by   diagrams   if  necessary,    why   the   images 
behave  as  stated  in  §  58  when  the  sextant  is  moved  in  the 
various  ways  there  described. 

7.  Show  that  the  limb  of  an  image  of  the  Sun  seen  in  a 
sextant   telescope,  which   is   preceding  with    respect    to   the 
apparent  motion  of  the  image,  corresponds  necessarily  to  the 
preceding   limb  of    the   Sun,    regardless   of    the    number   of 
reversals  to  which  said  image  may  have  been  subject  in  its 
progress  to  and  through  the  telescope.     By  the  use  of  this 
principle  show  that  when   observations  are  made  with   ap- 
proaching images  it  is  the   upper  limb  of  the  Sun  which  is 
being  observed  if  it  is  forenoon  and  the  lower  limb  if  it  is 
afternoon. 

8.  What  is  the  error  of  the  chronometer  on  local  mean 
time   from   the   last  half    of   the   set    of    observations  given 
in  §  63  ?  Ans.   —  2h  i im  538.5.* 

*  In  this  example  the  student  may  find  that  his  computation  gives  a 
result  differing  by  as  much  as  o8.2  from  the  one  here  given  on  account  of 
calling  o".5  a  whole  second,  where  it  has  in  this  computation  been  called 
zero,  or  vice  versa.  The  fact  that  such  a  difference  may  exist  may  be  used 
as  an  argument  for  carrying  the  computation  to  one  more  decimal  place. 
A  careful  investigation  indicates,  however,  that  such  a  procedure  would 
add  so  much  to  the  labor  of  computation,  especially  in  making  the  various 
interpolations,  that  it  is  not  considered  advisable.  If  the  computation  were 
carried  one  decimal  place  farther,  the  computed  result  from  a  complete  set 
of  observations  would  seldom  be  changed  by  more  than  o'.i,  whereas  the 
probable  error  of  that  result  is  ±  o*.3  or  ±  o'.4.  For  a  further  discussion 
of  the  question  of  the  number  of  decimal  places  to  which  a  computation 
should  be  carried,  see  §  277. 


96  GEODETIC  ASTRONOMY.  §  83. 

I 


CHAPTER    IV. 
THE    ASTRONOMICAL  TRANSIT. 

83.  THE  astronomical  transit  is  designed  primarily  to  be 
used  for  the  determination  of  time  with  its  telescope  in  the 
plane  of  the  meridian.  Its  essential  parts  are  a  telescope,  an 
axis  of  revolution  fixed  at  right  angles  to  the  telescope,  a 
suitable  support  for  said  axis,  such  that  it  shall  be  stable  in 
azimuth  and  inclination,  and  a  striding  level  with  which  the 
inclination  of  the  axis  may  be  determined. 

Fig.  10  shows  an  astronomical  transit  which  is  now  and 
has  been  for  several  years  past  in  use  in  the  U.  S.  Coast  and 
Geodetic  Survey  for  time  determinations  of  the  highest  order 
of  accuracy.  The  focal  length  (distance  from  the  lines  of  the 
eyepiece  diaphragm  to  the  optical  center  of  the  objective)  of 
the  telescope  AB  is  94  cm.  (37  in.).  The  clear  aperture  of 
the  object-glass  is  7.6  cm.  (3  in.),  and  the  magnifying  power 
with  the  diagonal  eyepiece,  A,  ordinarily  used  is  104 
diameters.  In  the  focus  of  the  eyepiece  is  a  thin  glass 
diaphragm  upon  which  are  ruled  lines  which  serve  the  same 
purpose  as  the  spider  lines  or  cross  wires  more  commonly 
placed  in  that  position  in  a  telescope.  The  system  of  lines 
consists  of  two  horizontal  lines  near  the  middle  of  the  field, 
and  thirteen  vertical  lines.  The  milled  head  shown  at  C  con- 
trols, by  means  of  a  rack  and  pinion,  the  distance  of  the 
diaphragm  from  the  object-glass,  and  serves  therefore  to  focus 
the  object-glass, — or,  in  other  words,  to  bring  the  image 


§  83.  DESCRIPTION   OF   TRANSIT.  97 

formed  by  the  object-glass  into  coincidence  with  the  dia- 
phragm. The  diaphragm,  and  the  corresponding  image 
formed  by  the  object-glass,  are  much  larger  than  the  field  of 
view  of  the  eyepiece.  To  enable  the  observer  to  see  various 
parts  of  the  diaphragm,  and  the  corresponding  portions  of 
the  image,  the  whole  eyepiece  proper  is  mounted  upon 
a  horizontal  slide  controlled  by  the  milled  head  shown  at  D. 

The  lines  of  the  diaphragm  are  seen  black  against  a  light 
field.  The  illumination  of  the  field  at  night  is  obtained  from 
one  of  the  lamps  shown  at  E.  The  light  from  the  lamp 
passes  in  through  the  perforated  end  of  the  horizontal  axis, 
and  is  reflected  down  to  the  eyepiece  by  a  small  mirror  in  the 
interior  of  the  telescope,  fastened  to  a  spindle  of  which  the 
milled  head  G  is  the  outer  end.  The  perforated  disk  shown 
at  F  carries  plain,  ground,  and  colored  glasses,  to  be  used  by 
the  observer  to  temper  the  illumination. 

The  horizontal  axis  FF  is  51.5  cm.  (2oJ  in.)  long,  ending 
in  pivots  of  bell  metal.  HH  is  the  striding  level,  in  position, 
resting  upon  the  pivots  of  the  horizontal  axis. 

The  iron  sub-base,  a  portion  of  which  shows  at  /,  is 
cemented  firmly  to  the  pier.  The  transit  base  is  carried  by 
three  foot-screws  resting  upon  this  sub-base.  The  device 
shown  at  J  serves  to  give  the  instrument  a  slow  motion  in 
azimuth. 

The  lever  K  actuates  a  cam  to  raise  the  cross-piece  Ly 
and  with  it  the  columns  MM.  The  horizontal  axis  is  then 
raised  sufficiently  upon  the  forks  at  the  upper  end  of  M  and 
M  to  clear  the  Ys.  The  cross-piece  L  is  then  free  to  turn 
(180°)  until  arrested  by  the  fixed  stops,  and  thus  to  reverse 
the  horizontal  axis  FF  in  the  Ys. 

The  setting  circles  NN  are  10  cm.  (4  in.)  in  diameter, 
are  graduated  to  20'  spaces,  and  are  read  to  single  minutes 
by  verniers.  They  are  set  to  read  zenith  distances. 


98  GEODETIC  ASTRONOMY.  §  84. 

Fig.  1 1  shows  another  type  of  transit  in  use  in  the 
U.  S.  C.  &  G.  S.  Its  peculiarities  are  the  folding  frame,  the 
graduated  scale  at  Q  to  facilitate  putting  the  telescope  in  the 
meridian,  and  the  fact  that  the  eyepiece  is  furnished  with  a 
movable  line  carried  by  a  micrometer  screw,  while  one  of  the 
setting  circles  carries  a  sensitive  level  so  that  the  instrument 
may  be  used  both  as  a  zenith  telescope  (see  Chapter  V)  and 
a  transit.  The  screw  Amoves  the  upper  base  55  in  azimuth 
with  respect  to  the  lower  base  RR. 

The  Theory  of  the  Transit. 

84.  If  a  transit  were  in  perfect  adjustment  the  line  of 
collimation  *  of  the  telescope  as  denned  by  the  mean  line  of 
the  reticle  would  be  at  right  angles  to  the  transverse  axis  upon 
which  it  revolves,  and  that  transverse  axis  would  be  in  the 
prime  vertical,  and  horizontal.  Under  these  circumstances 
the  line  of  collimation  would  always  lie  in  the  meridian  plane 

*  The  line  of  collimation  of  a  telescope  is  that  line  of  sight  to  which  all 
observations  are  referred.  In  an  engineer's  transit  the  line  of  collimation 
is  the  line  of  sight  on  which  all  observations  are  made,  and  is  defined  by  a 
vertical  line  in  the  middle  of  the  field  of  view  of  the  telescope.  In  an 
astronomical  transit  the  observations  are  made  on  the  several  lines  of  sight 
defined  by  the  several  lines  of  the  reticle.  These  various  observations  are 
all  referred  to  an  imaginary  line  of  sight,  or  line  of  collimation,  which  is 
defined,  however,  by  the  mean  of  all  the  lines,  and  not  by  the  middle  line. 
The  mean  line  is,  of  course,  near  to  the  middle  line,  the  spacing  of  the 
lines  in  the  reticle  being  made  as  nearly  uniform  as  possible. 

Imagine  a  plane  passed  through  any  line  of  the  reticle  of  a  telescope 
and  through  the  center  of  the  object-glass.  Imagine  the  plane  produced 
indefinitely  beyond  the  object-glass.  Evidently  every  point  of  which  the 
image  is  seen  in  the  telescope  in  apparent  coincidence  with  this  line  of  the 
reticle  must  lie  in  this  plane  in  space.  The  line  of  the  reticle  may  be  said 
to  define  this  plane  in  space,  or  a  point  of  the  reticle  line  may  be  said  to 
define  a  line  in  space.  For  convenience  a  line  of  the  reticle  is  ordinarily 
spoken  of  as  defining  a  line  of  sight  rather  than  a  plane  of  sight,  it  being 
tacitly  understood  that  one  point  only  of  the  reticle  line  is  referred  to — 
ordinarily  the  middle  point. 


§  85.  ADJUSTMENTS.  99 

and  the  local  sidereal  time  at  which  any  star  might  be  seen  in 
the  line  of  collimation  of  the  telescope  would  necessarily  be 
the  same  as  the  right  ascension  of  that  star.  In  observing 
meridian  transits  for  the  determination  of  time,  these  condi- 
tions are,  by  careful  adjustment  of  the  instrument,  fulfilled 
as  nearly  as  possible.  The  time  observations  themselves, 
and  certain  auxiliary  observations,  are  then  made  to  furnish 
determinations  of  the  errors  of  adjustment;  and  the  observed 
times  of  transit  are  corrected  as  nearly  as  may  be  to  what  they 
would  have  been  if  the  observations  had  been  made  with  a 
perfectly  adjusted  instrument.  The  observed  time  of  transit 
of  any  star,  as  thus  corrected,  minus  the  right  ascension  of  the 
star,  is  the  error  (on  local  sidereal  time)  of  the  chronometer 
with  which  the  observation  was  made. 

Adjustments  of  the  Transit. 

85.  Let  it  be  supposed  that  observations  are  about  to  be 
commenced  at  a  new  station  at  which  the  pier  and  shelter  for 
the  transit  have  been  prepared.  By  daylight  make  the  fol- 
lowing preparations  for  the  work  of  the  night. 

By  whatever  means  are  at  your  disposal  determine  the 
direction  of  the  meridian,  mark  it  upon  the  top  of  the  pier, 
and  put  the  foot-plates  of  the  transit  in  such  positions  that 
the  transit  telescope  will  swing  nearly  in  the  meridian  (true, 
not  magnetic).  A  compass  needle  will  serve  for  this  purpose 
if  no  other  more  accurate  and  equally  convenient  means  is  at 
hand.  An  accurate  determination  of  the  meridian  is  not  yet 
needed.  To  give  the  foot-plates  a  good  bearing  upon  the 
pier  and  to  fix  them  rigidly  in  position,  it  is  well  to  cement 
them  in  position  with  plaster  of  Paris. 

Set  up  the  transit  and  inspect  it.  Focus  the  telescope 
carefully  if  it  is  not  already  in  good  focus.  The  eyepiece 
may  be  first  focused  upon  the  reticle  with  the  telescope 


IOO  GEODETIC  ASTRONOMY,  §  86, 

turned  up  to  the  sky.  The  focus  for  most  distinct  vision  of 
the  reticle  lines  is  what  is  required.  Now  direct  the  telescope 
to  some  distant  object  (at  least  a  mile  away,  and  preferably 
much  farther)  and  focus  the  object-glass,  by  changing  its  dis- 
tance from  the  reticle,  so  that  when  the  eye  is  shifted  about 
in  front  of  the  eyepiece  there  is  no  apparent  change  of  rela- 
tive position  (or  parallax)  of  the  lines  of  the  reticle  and  of  the 
image  of  the  object.  If  the  eyepiece  itself  has  been  properly 
focused,  this  position  of  the  object-glass  will  also  be  the 
position  of  most  distinct  vision.  The  focus  pf  the  object-glass 
will  need  to  be  inspected  again  at  night,  and  corrected  if 
necessary,  using  a  star  as  the  object.  None  but  the  brightest 
stars  will  be  seen  at  all,  unless  the  focus  is  nearly  right. 

Bisect  some  well-defined  distant  object,  using  the  apparent 
upper  part  of  the  middle  vertical  line  of  the  reticle.  Rotate 
the  telescope  slightly  about  its  horizontal  axis  until  the  object 
is  seen  upon  the  apparent  lower  part  of  this  same  line.  If  the 
bisection  is  still  perfect,  no  adjustment  is  needed.  If,  how- 
ever, the  bisection  is  no  longer  perfect,  the  reticle  must  be 
rotated  about  the  axis  of  figure  of  the  telescope  until  the  line 
is  in  such  a  position  that  this  test  fails  to  discover  any  error. 

86.  Now  bisect  the  distant  object  with  the  middle  line  of 
the  reticle.  Reverse  the  telescope  axis  in  its  Ys.  If  the 
bisection  still  remains  perfect,  the  line  of  sight  defined  by  the 
middle  line  of  the  reticle  is  at  right  angles  to  the  horizontal 
axis  and  the  mean  line  may  be  assumed  to  be  sufficiently  near 
to  that  position.  If  necessary,  however,  make  the  adjust- 
ment by  moving  the  reticle  sidewise,  so  as  to  make  the  error 
of  collimation  small.  By  error  of  collimation  is  meant  the 
angle  between  the  line  of  sight  defined  by  the  mean  line  of 
the  reticle  and  a  plane  perpendicular  to  the  horizontal  axis  of 
the  telescope. 

Level  the  horizontal  axis  of  the  telescope.     Adjust  the 


§  8/.  ADJUSTMENTS.  IOI 

level  so  that  when  it  is  reversed  its  reading  will  change  but 
little. 

Test  the  finder  circles  to  see  that  they  have  no  index 
error.  Point  upon  an  object  and  read  one  of  the  finder 
circles.  Reverse  the  telescope,  point  upon  the  object  again, 
and  read  the  same  finder  circle  as  before.  The  mean  of  these 
two  readings  is  evidently  the  zenith  distance  of  the  object  (if 
the  circle  is  graduated  to  read  zenith  distances)  and  their  half 
difference  is  the  index  error  of  the  circle.  This  index  error 
may  be  made  zero  by  raising  or  lowering  one  end  or  the  other 
of  the  level  attached  to  the  vernier  of  the  finder  circle.  This 
same  process  will  evidently  serve  if  the  circle  reads  elevations 
instead  of  zenith  distances.  If  the  circle  is  to  be  made  to  read 
declinations  directly,  the  same  process  is  still  applicable. 
For  if  the  circle  be  made  to  read  zenith  distances  with  an 
index  error  equal  to  the  latitude  of  the  station,  its  readings 
will  be  declinations  for  one  position  of  the  telescope  (though 
not  for  the  other,  after  reversal).  The  other  circle  may  be 
made  to  read  declinations  for  the  other  position  of  the  tele- 
scope. 

A  "  finder  list"  of  stars,  showing  for  each  star  to  be 
observed  its  name,  magnitude,  setting  of  finder  circle,  and 
the  right  ascension  or  the  chronometer  time  of  transit  to  the 
nearest  minute,  will  be  found  to  be  a  convenience  in  the  night 
work.  In  making  out  a  finder  list  the  refraction  may  be 
neglected,  not  being  sufficient  to  throw  a  star  out  of  the  field 
of  the  telescope.  The  zenith  distance  of  a  star  is  then  0  —  tf, 
south  zenith  distances  being  reckoned  as  positive. 

The  Azimuth  Adjustment. 

87.  In  the  evening,  before  the  regular  observations  are 
commenced,  it  will  be  necessary  to  put  the  telescope,  more 
accurately  in  the  meridian.  Having  estimated  the  error  of 


IO2  GEODETIC  ASTRONOMY.  §  87. 

the  chronometer  in  any  available  way,  within  say  five  minutes, 
and  having  carefully  levelled  up  the  axis,  set  the  telescope  for 
some  bright  star  which  is  about  to  transit  within  10°  (say)  of 
the  zenith.  Observe  the  chronometer  time  of  transit  of  the 
star.  This  star  at  transit  being  nearly  in  the  zenith,  its  time 
of  transit  will  be  but  little  affected  by  the  azimuth  error  of 
the  instrument.  The  collimation  error  and  level  error  have 
been  made  small  by  adjustment.  Therefore  the  difference 
between  the  right  ascension  of  the  star  and  its  chronometer 
time  of  transit  will  be  a  close  approximation  to  the  error  of 
the  chronometer.  Now  set  the  telescope  for  some  slow- 
moving  star  which  will  transit  well  to  the  northward  of  the 
zenith  (let  us  say,  of  declination  greater  than  60°  and  a  north 
zenith  distance  of  more  than  20°).  Compute  its  chronometer 
time  of  transit,  using  the  approximate  chronometer  error  just 
obtained.  As  that  time  approaches  bisect  the  star  with  the 
middle  line  of  the  reticle,  and  keep  it  bisected,  following  the 
motion  of  the  star  in  azimuth  by  the  use  of  whatever  means 
have  been  furnished  on  that  particular  transit  for  that  purpose. 
Keep  the  bisection  perfect  till  the  chronometer  indicates  that 
the  star  is  on  the  meridian.  The  telescope  is  now  approxi- 
mately in  the  meridian. 

The  adjustment  may  be  tested  by  repeating  the  process, 
i.e.,  by  obtaining  a  closer  approximation  to  the  chronometer 
error  by  observing  another  star  near  the  zenith,  and  then 
comparing  the  computed  chronometer  time  of  transit  of  a 
slow  moving  northern  star  with  the  observed  chronometer 
time  of  its  transit.  If  the  star  transits,  apparently,  too  late, 
the  object-glass  is  too  far  west  (for  a  star  above  the  pole), 
and  vice  versa.  The  slow-motion  azimuth-screw  may  then  be 
used  to  reduce  the  azimuth  error.  This  process  of  reducing 
the  azimuth  error  will  be  much  more  rapid  and  certain,  if 
instead  of  simply  guessing  at  the  amount  of  movement  which 


§  88.  DIRECTIONS  FOR   OBSERVING.  10$ 

must  be  given  to  the  azimuth-screw,  one  computes  roughly 
what  fraction  of  a  turn  must  be  given  to  it.  This  may  be 
done  by  computing  the  azimuth  error  of  the  instrument 
roughly  by  the  method  indicated  in  §  102,  having  previously 
determined  the  value  of  one  turn  of  the  screw.  An  experi- 
enced observer  will  usually  be  able  on  the  second  or  third 
trial  to  reduce  the  azimuth  error  to  less  than  is(=  15"). 

The  table  given  in  §  310  will  be  found  convenient  in 
making  the  first  approximation  to  the  meridian. 

Directions  for  Observing. 

88.  The  instrument  being  completely  adjusted  and  the 
axis  levelled,  set  the  telescope  for  the  first  star.  It  is  not 
advisable  to  use  the  horizontal  axis  clamp  during  observa- 
tions, for  its  action  may  have  a  slight  tendency  to  raise  one 
end  or  the  other  of  the  axis.  See  to  it,  loading  one  end  if 
necessary,  that  the  centre  of  gravity  of  the  telescope  is  at  its 
horizontal  axis,  and  then  depend  upon  the  friction  at  the 
pivots  to  keep  the  telescope  in  whatever  position  it  is  placed. 
When  the  star  enters  the  field,  bring  it  between  the  horizon- 
tal lines  of  the  reticle,  if  it  is  not  already  there,  by  rapping 
the  telescope  lightly.  Center  the  eyepiece  so  that  the 
vertical  line  nearest  the  star  is  in  the  apparent  middle  of  the 
field  of  view.  As  the  star  approaches  the  line  pick  up  the 
beat  of  the  chronometer.*  Observe  the  chronometer  time 
of  transit  across  the  line,  estimating  to  tenths  of  seconds. 
Then  center,  the  eyepiece  on  the  second  line  and  observe 
the  transit  there,  and  so  on,  until  observations  have  been 
made  upon  all  the  lines,  taking  care  always  to  keep  the  eye- 
piece centered  upon  the  line  which  is  in  use. 

The  directions  and  suggestions  given  in  §  60  for  observing 
time  apply  here  with  equal  force.  In  order  to  estimate 

*  The  eye  and  ear  method  of  observing  without  a  chronograph  is  here 
referred  to.  For  a  description  of  the  chronograph  and  the  method  of  using 
it  in  observing,  see  §  89. 


104  GEODETIC  ASTRONOMY.  §  88. 

tenths  of  seconds  it  is  necessary  to  divide  the  half-second 
interval  given  directly  by  the  chronometer  into  fifths  by 
some  mental  process.  To  secure  accuracy  and  ease  in  mak- 
ing this  estimate  it  is  advisable  to  transform  it  into  a  proc- 
ess of  estimating  relative  distances.  The  apparent  motion 
of  the  star  image  is  nearly  uniform  (not  quite  so  on  account 
of  disturbance  by  irregular  refraction).  Let  us  suppose  that 
at  the  instant  when  the  chronometer  tick  which  indicates  the 
time  to  be  Is.  5  is  heard  the  star  is  seen  at  A  (Fig.  12).  Let 
the  observer  retain  a  mental  picture  of  this  relative  position 
of  the  star  and  the  line.  When  the  chronometer  tick  for  2s. o 
is  heard,  he  sees  the  star  at  B.  If  he  has  retained  (for  os.  5 
only)  the  mental  picture  referred  to  above,  he  has  before  his 
mind's  eye  exactly  what  is  indicated  in  the  figure.  He  esti- 
mates the  ratio  of  the  distances  from  A  to  the  line  and  from 
A  to  B,  and  concludes  that  the  ratio  is  nearer  to  f  than  to  £, 
or  f,  and  calls  the  time  of  transit  of  the  star  across  line  /,  i1.?. 
Though  this  mental  process  may  seem  awkward  at  first,  it 
will  ultimately  be  found  to  be  both  easier  and  more  accurate 
than  the  direct  process,  for  all  cases  in  which  the  star  has  an 
apparent  motion  which  is  sufficiently  rapid  to  make  the  dis- 
tance AB  appreciable  in  a  half-second.  An  experienced 
observer,  using  this  process,  is  able  to  estimate  the  time  of 
transit  of  a  star's  image  across  each  line  of  the  reticle  with  a 
probable  error  of  about  ±  o8.  i . 

It  is  well  here  to  bear  in  mind  the  suggestion  given 
in  §  60,  that  he  who  hesitates  is  inaccurate.  The  successful 
observer  decides  promptly,  but  without  hurry,  upon  the 
second  and  tenth  at  which  the  transit  occurred. 

At  convenient  intervals  between  stars  the  striding  level 
should  be  read  in  each  of  its  positions  upon  the  horizontal 
axis.  At  about  the  middle  of  the  observations  which  are  to 
constitute  a  set  the  telescope  should  be  reversed,  so  that  the 
effect  of  the  error  of  collimation  (and  inequality  of  pivots) 


§  89.  THE   CHRONOGRAPH.  10$ 

upon  the  apparent  time  of  transit  may  be  reversed  in  sign. 
Each  half-set  should  contain  one  slow-moving  star  (of  large 
declination)  to  furnish  a  good  determination  of  the  azimuth 
error  of  the  instrument. 

The  telegraphic  longitude  parties  of  the  Coast  and 
Geodetic  Survey  make  the  best  time  determinations  that  are 
made  with  portable  instruments  at  present  in  this  country. 
In  their  practice  ten  stars  are  observed  in  each  set,  five  before 
and  five  after  reversal  of  the  telescope.  From  two  to  four 
readings  of  the  level  are  taken  in  each  of  its  positions  in  each 
half-set.  Care  is  taken  to  have  the  telescope  at  different 
inclinations  during  the  different  readings  of  the  level,  inclined 
sometimes  to  the  south  and  sometimes  to  the  north,  so  that 
the  level  may  rest  in  turn  upon  various  parts  of  the  pivots. 
This  is  done  to  eliminate,  in  part  at  least,  the  effect  of 
irregularity  in  the  figure  of  the  pivots  upon  the  determination 
of  the  inclination  of  the  axis. 

The  Chronograph. 

89.  The  preceding  directions  for  observing  were  given  on 
the  supposition  that  the  eye  and  ear  method  of  observing  the 
times  of  transit  is  to  be  used.  If,  instead,  the  time  obser- 
vation proper  is  made  with  a  chronograph,  the  method  is 
changed  in  that  one  particular  only. 

A  common  form  of  the  chronograph  is  shown  in  Fig.  13. 
The  train  of  gear-wheels  partially  visible  through  the  back 
glass  of  the  case  at  F  is  driven  by  a  falling  weight,  and  drives 
the  speed  governor  at  AECCDD,  the  screw  /,  and  the 
cylinder  H.  As  the  speed  of  rotation  of  the  governor 
increases,  the  weights  CC  move  farther  from  the  axis  until  a 
small  projection  on  one  of  them  strikes  the  hook  at  E  and 
carries  it  along.  This  hook  carries  with  it  in  its  rotation  the 
small  weight  A.  The  result  of  the  impact  and  of  the  added 
friction  at  the  base  of  A  is  to  cause  the  speed  of  the  governor 


IO6  GEODETIC  ASTRONOMY.  §  89. 

to  decrease  until  the  hook  E  is  released.  The  speed  then 
increases  until  the  hook  is  engaged,  decreases  again  until  it  is 
released,  and  so  on.  The  total  range  of  variation  in  the  speed 
is,  however,  surprisingly  small, — so  small  that  in  interpreting 
the  record  of  the  chronograph  the  speed  is  assumed  to  be 
uniform  during  the  intervals  between  clock  breaks.  By 
moving  the  adjusting  nuts  DD  upon  their  screws,  the  critical 
speed  at  which  the  hook  is  engaged  may  be  adjusted.  The 
carriage  M  is  moved  parallel  to  the  axis  of"  the  cylinder  H  by 
the  screw  /.  The  pen  G,  carried  by  an  arm  projecting  from 
the  carriage  M,  tends  to  trace  a  helix  at  a  uniform  rate  upon 
the  paper,  or  "  chronograph  sheet,"  stretched  upon  the 
cylinder.  The  magnets  KK  are  in  an  electric  circuit  (through 
the  wires  Z),  with  a  break-circuit  clock  or  chronometer. 
Whenever  the  circuit  is  broken,  the  armature  A7"  is  released  and 
the  back  portion  of  the  arm  carrying  the  pen  is  drawn  back, 
by  a  spring,  to  contact  with  the  stop  at  J.  The  pen  then 
makes  a  small  offset  from  the  helix.  It  returns  to  the  helix 
as  soon  as  the  current  is  renewed.  As  a  result  the  equal  in- 
tervals of  time  between  the  instants  at  which  the  chronometer 
breaks  the  electric  circuit  are  indicated  by  equal  linear  in- 
tervals between  offsets  on  the  line  drawn  by  the  pen,  the 
speed  being  kept  constant  by  the  governor.  The  chronome- 
ter is  usually  arranged  to  break  the  circuit  every  second  or 
every  alternate  second,  and  to  indicate  the  beginning  of  each 
minute  by  omitting  one  break.  The  hours  and  minutes  may 
be  identified  by  recording  at  some  point  upon  the  sheet  the 
corresponding  reading  of  the  face  of  the  chronometer. 

The  electric  circuit  passing  through  the  magnets  KK^  the 
chronometer,  and  battery  also  passes  through  a  break-circuit 
key  in  the  hand  of  the  observer.  To  record  the  exact  time 
of  occurrence  of  any  phenomenon  he  presses  the  key  at  that 
instant,  breaks  the  circuit,  and  produces  an  additional  offset 
in  the  helix,  of  which  the  position  indicates  accurately  the 


§  go.  RECORD   AND    COMPUTATION.  IO? 

time  at  which  it  was  made.  In  observing  a  star,  to  determine 
a  chronometer  error  the  instant  of  transit  of  the  star  image 
across  each  line  of  the  reticle  is  so  recorded.  To  read  the 
fractions  of  seconds  from  the  chronograph  sheet  it  is  con- 
venient to  use  a  scale  divided  into  intervals  corresponding  to 
tenths  of  seconds.  The  process  of  reading  is  also  facilitated 
by  writing  the  number  of  the  second  at  the  head  of  each  col- 
umn, and  of  the  minute  on  each  line,  of  the  chronograph 
sheet  before  beginning  to  read. 

The  experienced  observer  gains  very  little  in  accuracy  by 
substituting  the  chronographic  method  of  observing  time  for 
the  eye  and  ear  method.  He  gains  somewhat  in  the  con- 
venience and  rapidity  with  which  his  night's  record  is  made. 
These  small  gains  are  not  usually  sufficient  to  justify  the  use 
of  the  chronograph  in  the  field,  except  in  connection  with 
telegraphic  determinations  of  longitude.  In  that  case  the 
chronographic  method  has  special  advantages  (see  §§  234-242). 

Example  of  Record  and  Computation. 

90.  The  following  time  set  was  observed  as  a  part  of  the 
telegraphic  longitude  work  of  the  Coast  and  Geodetic  Survey 
in  May,  1896.  The  observations  were  made  with  a  chrono- 
graph. The  explanation  of  each  separate  portion  of  the 
computation  is  given  in  detail  under  the  appropriate  heading 
in  the  following  sections. 

The  constants  of  the  instrument  used  in  this  example  are 
here  inserted  for  convenient  reference. 

One  division  of  the  striding  level  =  i".674. 

Pivot  inequality  =  —  O8.oio  with  band  west. 

Equatorial  intervals  of  lines  with  band  west : 

Line  i.  —  15". 20  Lines.  —  2s. 52  Line  9.  -)- IO"-°9 

"     2.  —  i2s.69  "     6.  +  o8.09  "   10.  +  i28.65 

"     3.  —  ios.i5  "     7.  +28.52  "   ii.  4-158.15 

"     4.  —    58.o6  "     8.  +  5s.  1 1 


IO8  GEODETIC  ASTRONOMY.  §  Q2. 

Time  of  Transit  Across  Mean  Line. 

92.  If  the  transit  of  the  star  across  every  line  of  the  reticle 
is  observed,  the  time  of  transit  across  the  mean  line,  or  line  of 
collimation,  is  evidently  obtained  by  taking  the  mean  of  the 
several  observed  times.  In  obtaining  the  sum  of  the  several 
times  for  this  purpose  an  error  of  a  whole  second  in  any  one 
observed  time,  which  might  otherwise  remain  unnoticed,  will 
be  detected  by  the  use  of  the  auxiliary  sums  shown  in  the 
little  column  just  after  the  observed  times,  namely,  the  sum 
of  the  first  and  last  times,  of  the  second  and  last  but  one, 
third  and  last  but  two,  etc.  These  auxiliary  sums  should  be 
nearly  the  same  and  nearly  equal  to  double  the  time  on  the 
middle  line.  The  unexpressed  minute  for  each  is  the  same 
as  that  for  the  middle  line.  The  sum  of  these  auxiliary  sums 
and  of  the  middle  time  is  the  total  sum  required  in  comput- 
ing the  mean. 

It  will  frequently  happen,  especially  on  partially  cloudy 
or  hazy  nights,  that  the  transits  of  a  star  across  several  lines 
of  the  reticle  will  be  successfully  observed,  and  yet  the 
observer  may  fail  utterly  to  secure  the  transits  across  the 
remaining  lines.  It  then  becomes  necessary  to  reduce  the 
mean  of  the  observed  times  of  transit  across  certain  of  the 
lines  to  the  mean  of  all  of  the  lines. 

Let  t^  /„,  /8,  .  .  .  be  the  observed  times  of  transit  across 
the  successive  lines,  and  let  tm  be  their  mean,  or  the  time  of 
transit  across  the  mean  line. 

Let  z,,  za,  z,,  ...  be  the  equatorial  intervals  of  the  succes- 
sive lines  from  the  mean  line,  or  the  intervals  of  time  which 
elapse  for  an  equatorial  star  (star  of  zero  declination)  between 
transits  across  the  separate  lines  and  the  transit  across  the 
mean  line. 

Then    for  an    equatorial  star    ^  =  t^  —  tm,    *3  =  /3  —  tm> 

*'•   =   *,   —    *m>    •    •    • 


§  92.  RED  UCTION   TO   MEAN  LINE.  109 

To  determine  the  relation  for  any  other  star  between  inj 
the  equatorial  interval  for  any  line,  and  tn  —  tm  (in  which  tn 
is  the  time  of  transit  over  that  line),  deal  with  the  spherical 
triangle  defined  by  the  pole,  the  star  at  the  instant  when  it  is 
on  any  line,  and  the  star  at  the  instant  when  it  is  on  the  mean 
line.  In  Fig.  14,  let  P,  A,  and  B  represent  these  points 
respectively.  The  sides  PB  and  PA  are  the  polar  distance  of 
the  star  (—  90°  —  #).  The  angle  at  P  expressed  in  seconds 
of  arc  is  i$(tH  —  tm},  (tn  —  tm)  being  expressed  in  seconds  of 
time.  The  side  AB,  expressed  in  seconds  of  arc,  may  be 
taken  equal  to  15^,  in  being  expressed  in  seconds  of  time. 
Using  the  law  that  the  sines  of  the  sides  of  any  spherical  tri- 
angle are  proportional  to  the  sines  of  the  opposite  angles, 
there  is  obtained 

sin  154  :  sin  (90°  —  d)  =  sin  i$(tn  —  Q  :  sin  A.       (19) 

The  arc  AB  corresponding  to  the  equatorial  interval  for  a 
line  will  seldom  exceed  15'  in  any  transit,  and  is  usually  much 
less.  For  such  an  isosceles  spherical  triangle  as  this,  AB 
being  short,  angles  A  and  B  are  necessarily  nearly  equal  to 
90°.  Assuming  as  an  approximation  that  A  =  90°,  (19)  may 
be  written  sin  154  —  sin  i$(tn  —  tm)  cos  d.  Again,  assuming 
that  the  small  angles  i$iH  and  i$(tn  —  tm)  are  proportional  to 
their  sines,  and  dividing  both  members  by  15,  we  obtain 

4  =  (^  —  O  cos  $ (20) 

In  the  derivation  of  (20)  besides  the  approximations  men- 
tioned, there  is  another  in  the  assumption  that  AB  corre- 
sponds to  the  perpendicular  distance  between  the  two  lines  of 
the  reticle  concerned,  whereas  none  but  images  of  equatorial 
stars  will  pursue  a  path  across  the  reticle  which  is  perpendic- 
ular to  all  the  lines.  Every  other  star  image  follows  an 


HO  GEODETIC  ASTRONOMY.  §93. 

apparent  path  which  is  curved  (the  apparent  radius  of  curva- 
ture being  less,  the  greater  the  declination),  and  therefore  not 
perpendicular  to  more  than  one  line  of  the  reticle.  So  AB 
corresponds,  except  for  an  equatorial  star,  to  an  oblique  dis- 
tance between  lines  of  the  reticle,  the  obliquity  being  exceed- 
ingly small.  In  spite  of  all  these  approximations  it  has  been 
found  by  comparing  (20)  with  the  formula  expressing  the 
exact  relation  between  in  and  (tn—tm)  that  for  an  extreme 
value  of  in  =  6o8  the  computed  value  of  (tn  —  t^)  will  be  in 
error  by  less  than  O8.oi  for  any  star  of  declination  less  than 
70°,  and  that  for  a  star  of  declination  85°  the  error  is  only 
08.3.* 

93.  Let  us  deal  now  with  such  a  case  as  that  of  star  17 
H.  Can.  Ven.  in  the  preceding  computation,  in  which  the  star 
image  was  observed  to  transit  across  the  first  ten  of  the  eleven 
lines  of  the  reticle  and  the  transit  across  the  eleventh  line  was 
missed.  Suppose  that  the  equatorial  intervals  *,,  za,  z,,  .  .  . 
have  previously  been  determined  by  special  observations  as 
indicated  in  §  114. 

From  (20)  we  may  write 


sec 
sec 
sec 


(21) 


*m  =  *io  —  *io  sec  tf 
whence 

(*,+vH+  •  •  •  +*'..)  sec 


o 


*  For  the  exact  treatment  of  this  problem,  see  Chauvenet's  Astronomy, 
vol.  II.  pp.  146-149;  or  Doolittle's  Astronomy,  pp.  291-293. 


§93-  REDUCTION   TO  MEAN  LINE.        .  Ill 

in  which  all  quantities  are  known.  By  the  use  of  (22)  the 
time  of  transit  tm  across  the  mean  line  may  be  computed 
even  though  the  transits  across  certain  of  the  lines  are 
missed. 

Adding  the  corresponding  terms  of  the  separate  equations 
of  (21),  together  with  the  equation  tm  =  /„  —  tn  sec  6  (which 
is  true  though  tn  is  unknown),  and  dividing  by  1  1,  there  is 
obtained 


sec  <y 

-•  (23) 


But  tm  is  by  definition  *»  +  *•  +  *«  +  •  •  '  +A\      Therefore 

(23)  becomes 

o  =  —  ft  +  *;  +  f,  +  .  .  .  +  in)  sec  tf, 
and 

-  ft  +  *.  +  *,+  •••  +  *,.)  sec  <y  =  (*'„)  sec  tf  : 
(22)  may  now  also  be  written  in  the  form 

,         **  +  *.+  *.+  .  •  •+*...    ft.)  sec* 

/-=  ~T^~  -15—  • 

For  the  particular  case  in  hand  of  the  imperfect  transit  of 
17  H.  Can.  Ven.,  sec  6  =  C  =  1.26,  in  =  +  15".  15,  and  (24) 
becomes 


1 1 2  GEODE  TIC  A  S  TRONOM  Y.  §  94. 

For  the  general  case  (22)  and  (24)  may  be  written,  respec- 
tively, 

(sum  of  equatorial  intervals 
of  observed  lines)  (sec  d) 

tm  —  mean  of  observed  times - — —    — • \— T-. ;   .     (25) 

number  of  observed  lines 

(sum  of  equatorial  intervals 
of  missed  lines)  (sec  d) 

tm  =  mean  of  observed  times  4- r 7 — r r~r- — ' — •   •     (26) 

number  of  observed  lines 

The  first  or  second  of  these  two  formulae  is  to  be  used, 
respectively,  according  to  whether  less  than  half  or  more  than 
half  of  the  lines  were  observed. 

Inclination  Correction. 

94.  In  the  following  treatment  it  will  be  assumed  that  the 
two  pivots  upon  which  the  telescope  turns  are  circular 
cylinders  having  nearly,  but  not  exactly,  equal  radii,  and  that 
the  two  inverted  Ys  forming  a  part  of  the  striding  level,  and 
the  two  Ys  in  which  the  telescope  pivots  rest,  all  have  equal 
angles. 

If  Fig.  15  represent  a  cross-section  through  one  of  the 
pivots  perpendicular  to  its  axis,  it  is  assumed  that  the  curve 
GEBD  is  a  perfect  circle,  of  which  the  center  is  at  C,  and  that 
the  angles  GFE  and  DAB  are  equal  to  each  other  and  to  the 
corresponding  angles  at  the  other  pivot.  With  these  assump- 
tions the  distances  FC  and  AC,  from  the  vertex  of  the  level 
Y  and  from  the  vertex  of  the  supporting  Y  to  the  center  of 
the  pivot,  are  equal. 

In  Fig.  1 6  let  CC'  be  the  line  joining  the  centers  of  the 
two  pivots,  the  axis  about  which  the  telescope  rotates,  of 
which  the  inclination  is  required.  Let  F  and  F'  be  the  ver- 
tices of  the  level  Ys,  and  A  and  A'  the  vertices  of  the  sup- 
porting Ys.  Suppose  the  radius  of  the  pivot  of  which  the 
centre  is  C'  is  greater  than  that  of  the  pivot  at  C.  Then 


§94-  INCLINATION   CORRECTION.  11$ 

the  distance  F'C'(=  A'C')  is  greater  than  the  distance 
FC(=  AC).  Draw  CG  parallel  to  FF'  and  CH  parallel  to 
AA'.  The  inclination  of  the  line  FF'  (equal  to  the  inclina- 
tion of  the  line  CG)  is  directly  measured  by  the  level  readings. 

The  angle  between  FF'  and  CC'  is  the  angle  GCC'l  = 

This  is  the  correction  to  be  applied  to  the  inclination  as  given 
directly  by  the  level  readings  to  obtain  the  inclination  of  the 
axis.  If  the  telescope  axis  were  reversed  in  the  Ys  the  line 
A  A'  joining  the  vertices  of  the  supporting  Ys  would  remain 
unchanged,  and  the  line  FF'  would  assume  a  new  position 
F"F"f  such  that  the  angle  between  FF'  and  F"F"'  is  four 
times  GCC\_=  2(GCHJ].  Let  ftw  and  ftB  designate  the 
inclination,  as  given  by  the  level  readings,  for  lamp  west  and 
lamp  east,  respectively  (using  the  position  of  the  lamp  which 
illuminates  the  interior  of  the  telescope  as  a  convenient 
means  of  designating  the  position  of  the  telescope),  and  bw 
and  be  the  corresponding  inclinations  of  the  axis  CC'.  Let 
the  angle  GCC'  be  called  piy  or  pivot  inequality.  Let  all 
inclinations  be  considered  positive  when  the  west  end  is 
higher  than  the  east.  Then 

*  f  i  —  8,,,-L-  A.-  } 

(27) 


and  pl=. (28) 

Let  w  and  e  be  the  readings  of  the  west  and  east  ends, 
respectively,  of  the  bubble  of  the  striding  level  for  a  given 

*  These  formulae  are  exact  only  in  case  the  angle  of  the  level  Ys  is  the 
same  as  the  angle  of  the  supporting  Ys.  For  ordinary  cases,  however,  in 
which  pi  is  small  and  the  Ys  have  angles  which  do  not  differ  greatly,  they 
are  sufficiently  exact.  For  the  full  treatment  of  this  problem  for  the  general 
case,  see  Chauvenet's  Astronomy,  vol.  11.  pp.  153-158. 


114  GEODETIC  ASTRONOMY.  §95. 

position  of  the  telescope  axis.  Let  w1  and  e'  be  the  corre- 
sponding west  and  east  readings  after  the  level  is  reversed, 
the  telescope  axis  remaining  as  it  was.  Let  d  be  the  value 
of  a  division  of  the  level  in  seconds  of  arc.  Then  for  ftt 
the  apparent  inclination  of  the  telescope  axis,  expressed  in 
seconds  of  time,  we  may  write,  if  the  level  divisions  are 
numbered  in  both  directions  from  the  middle, 


or  in  more  convenient  form  for  numerical  work, 

0=!(w  +  «0--('  +  O$g5.  •    •    •    (29) 

d 

in  which  ^-  is  a  constant  for  the  level. 
DO 

If  the  level  divisions  are  numbered  continuously  from  one 
end  of  the  level  to  the  other,  (29)  takes  the  form 

ft  =  {(«,  +  *)-(*  +  ,')}£;    .     .     .     (30) 

in  which  the  primed  letters  refer  to  that  position  of  level  in 
which  the  division  marked  zero  is  at  the  western  end. 

95.  It  still  remains  to  derive  the  relation  between  b,  the 
inclination  of  the  rotation  axis  of  the  telescope,  and  the  cor- 
rection to  the  observed  time  of  transit  of  a  star  to  reduce  it 
to  what  it  would  be  if  the  rotation  axis  were  truly  horizontal. 

If  the  error  of  collimation  were  zero  and  the  rotation  axis 
of  the  telescope  horizontal  and  in  the  prime  vertical,  the  line 
of  collimation  would  describe  the  meridian  upon  the  celestial 
sphere  when  the  telescope  was  rotated  upon  its  axis.  If  now 
the  other  errors,  of  azimuth  and  collimation,  be  assumed  to 
remain  zero  but  the  rotation  axis  is  moved  slightly  out  of  the 


§95-  INCLINATION  CORRECTION.  11$ 

horizontal  by  an  angle  b,  the  line  of  collimation  will  describe 
a  great  circle  intersecting  the  meridian  at  the  south  and  north 
points  of  the  horizon  and  making  an  angle  b  with  it  at  those 
points.  In  Fig.  17  let  B  be  the  position  occupied  by  a  star  at 
the  instant  when  it  would  be  observed  to  cross  the  mean  line 
of  a  transit  in  perfect  adjustment.  Let  B'  be  its  position 
when  it  would  be  observed  to  cross  the  mean  line  of  a  transit 
having  an  axis  inclination  b,  but  otherwise  in  perfect  adjust- 
ment. Let  5  be  the  south  point  of  the  horizon.  In  the 
spherical  triangle  BB'S  the  angle  5  in  seconds  of  arc  =  15^, 
SB  is  the  altitude  of  the  star  when  it  is  on  the  meridian, 
—  90°  —  (0  —  <?),-  and  the  angle  BB'S  is  almost  exactly  a 
right  angle.  From  the  law  of  proportionality  of  sines,  sin 
15^  :  sin  BB'  =  sin  BB'S  :  sin  {90°—  (0  —  $)}. 

Whence,  the  small  angles  i$b  and  BB'  being  assumed 
proportional  to  their  sines, 

BB'=i$b  cos  C (30 

Treating  now  the  spherical  triangle  defined  by  B,  B' ,  and 
the  pole,  just  as  the  spherical  triangle  in  figure  was  treated, 
it  may  be  shown  that  the  hour-angle  subtended  at  the  pole 
by  BB'  is  BB'  sec  6.  Substituting  this  in  (31)  and  reducing 
to  time,  there  is  obtained  as  the  hour-angle,  or  elapsed  inter- 
val of  time  (in  seconds)  between  the  position  B  and  the 
position  B'j 

b  cos  C  sec  d (32) 

This  is  the  required  correction  to  the  observed  time  of 
transit.  For  convenience  this  may  be  written 

Inclination  correction  =  Bb,        .     .      .     (33) 

in  which  B  —  cos  £  sec  #,  and  may  be  found  tabulated  for 
the  arguments  £  and  d  in  §  299. 


1 1 6  GEODE  TIC  A  S  TRONOM  V.  §  96. 

The  approximations  in  the  derivation  of  (32)  are  of  the 
same  order  as  those  made  in  deriving  (20).  If  4"  be  allowed 
as  a  maximum  value  for  b,  the  error  of  the  formula  will  be 
much  less  than  os.oi  for  any  star  of  declination  less  than  80°. 
Under  ordinary  circumstances  b  will  seldom  be  as  great  as  Is. 

In  deriving  ft  from  the  level  readings  it  is  sometimes 
assumed  that  the  inclination  is  variable,  and  that  each  set  of 
level  readings  gives  the  inclination  at  that  particular  time. 
Under  good  conditions,  however,  the  variation  of  the  inclina- 
tion during  any  half  set  is  probably  less  than  the  error  of  any 
one  determination  of  the  inclination.  It  is  advisable,  there- 
fore, to  assume  the  inclination  constant  during  a  half  set.  The 
method  of  computing  this  mean  inclination  from  the  level 
readings  is  sufficiently  shown  in  the  example  in  §  91.  A  dis- 
tinction is  made  between  level  readings  with  objective  north 
and  readings  with  objective  south  on  account  of  the  possibility 
that  the  level  readings  may  be  affected  by  irregularities  in  the 
shape  of  the  pivots. 

Correction  for  Diurnal  Aberration. 

96.  The  effect  of  the  annual  aberration,  due  to  the  motion 
of  the  Earth  in  its  orbit  (§  46),  is  taken  into  account  in  com- 
puting the  apparent  star  place.  But  the  effect  of  the  diurnal 
aberration,  due  to  the  rotation  of  the  Earth  on  its  axis,  must 
be  dealt  with  in  the  present  computation.  In  round  numbers 
the  velocity  of  light  is  186000  miles  per  second,  and  the  linear 
velocity  of  a  point  on  the  Earth's  equator  due  to  the  diurnal 
rotation  is  0.288  mile  per  second.  The  linear  velocity  of 
any  point  on  the  Earth  in  latitude  0  is  then  0.288  cos  0  mile 
per  second.  The  apparent  displacement  of  a  star  on  the 
meridian  is 

0.288   COS   0 

*=tan        .86000 


§97«         AZIMUTH  AND    COLLIMA  TION  CORRECTIONS.        1 1/ 

the  motion  of  the  observer  being  at  right  angles  to  the  line  of 
sight.  The  hour-angle  k,  corresponding  to  the  displacement 
k ' ,  is  found  by  applying  the  method  of  the  latter  part  of  §  92 
to  the  spherical  triangle  defined  by  the  true  position  of  the 
star,  its  displaced  position,  and  the  pole.  It  is  thus  found 
that 

,  0.288  cos  0 
*  =  *sec*  =  tan-      l86oQQ     sec  J. 

Keeping  in  mind  that  the  angles  k'  and  k  are  very  small, 
this  may  be  written 

0.288 

k  =  —^2 —         T>  cos  0  sec  #  =  o".3IQ  cos  0  sec  # 

186000  tan  i 

=  O8.02i  cos  0  sec  tf.    .     (35) 

For  convenience  k  is  tabulated  in  terms  of  0  and  d  in 
§  301. 

As  the  aberration  causes  the  star  to  appear  too  far  east, 
the  observed  time  of  transit  is  too  late,  and  k  is  negative  when 
applied  as  a  correction  to  the  observed  times  (except  for  sub- 
polars). 

Azimuth  Correction. 

97.  If  the  transit  is  otherwise  in  perfect  adjustment  but 
has  a  small  error  in  azimuth,  the  line  of  collimation  will 
describe  a  vertical  circle,  i.e.,  a  great  circle  passing  through 
the  zenith,  at  an  angle  with  the  meridian  (measured  at  the 
zenith)  which  we  will  call  a. 

In  Fig.  1 8,  let  z  be  the  zenith,  B  be  the  position  of  a  star 
when  it  is  on  the  meridian,  and  B'  its  position  when  observed 
crossing  the  line  of  collimation  of  a  transit  which  has  an 
azimuth  error  a.  By  applying  the  process  of  the  latter  part 
of  §  92  to  this  spherical  triangle  it  may  be  shown  that 
BB'  =  a  sin  . 


Il8  GEODETIC  ASTRONOMY.  §98. 

Applying  the  same  process  again  to  the  spherical  triangle 
defined  by  B,  B' ,  and  the  pole,  it  may  be  shown  that  the 
corresponding  hour-angle  is  BB'  sec  d. 

Let  the  angle  a  be  expressed  in  seconds  of  time,  and  be 
called  positive  when  the  object-glass  is  too  far  east  with  the 
telescope  pointing  southward.  Then  the  required  correction 
to  the  observed  times,  equal  to  the  time  elapsed  between 
position  B'  and  B  of  the  star,  may  be  written 

Azimuth  correction  =  a  sin  £  sec  d  =  Aa,       .     (36) 

in  which  A  is  written  for  sin  C,  sec  tf,  and  is  tabulated  in  terms 
of  £  and  d  in  §  299. 

The  methods  of  deriving  a  from  the  time  observations  will 
be  treated  later  (§§  100-110). 

Collimation  Correction. 

98.  If  the  instrument  is  otherwise  in  perfect  adjustment 
but  has  a  small  error  of  collimation  (§  86),  the  mean  line 
describes  a  small  circle  parallel  to  the  meridian,  at  an  angular 
distance  c,  the  error  of  collimation,  from  it,  when  the  telescope 
is  rotated  about  its  horizontal  axis.  By  the  same  line  of 
reasoning  that  was  used  in  §  92  in  dealing  with  the  intervals 
of  the  various  lines  from  the  mean  line,  it  may  be  shown  that 
if  c  be  expressed  in  time,  then  the 

Collimation  correction  *  =  c  sec  8  =  Cc,  .      .     (37) 

*  Objection  may  be  made  to  the  methods  used  in  deriving  the  formulae 
of  §§  92-98  because  of  the  many  approximations  involved  in  them.  For, 
in  addition  to  the  stated  approximations  that  have  been  made  in  the 
derivations,  the  fact  that  the  formulae  for  inclination,  azimuth,  and  col- 
limation are  not  independent  has  been  neglected,  and  each  treated  as  if 
entirely  independent  of  the  other. 

Is  such  objection  valid  ?  Our  present  purpose  is  to  furnish  the 
engineer  with  such  mathematical  formulas  (together  with  an  intelligent 


§99'  CORRECTION  FOR  RATE.  IIQ 

in  which  C  is  written  for  sec  8  and  is  tabulated  in  terms  of  £ 
in  §  299.  The  collimation  error  necessarily  changes  sign 
when  the  telescope  axis  is  reversed  in  its  Ys.  Let  c  be  the 
error  of  collimation  with  lamp  or  band  east,  and  let  it  be  called 
positive  when  the  star  (above  the  pole)  is  observed  too  soon. 
To  take  account  of  the  change  in  sign  of  the  collimation  error 
with  lamp  west,  let  the  sign  of  C  be  reversed  (so  that  C  —  — 
sec  6)  whenever  the  lamp  is  west.  With  this  convention  as 
to  the  sign  of  C  the  algebraic  sign  of  the  product  Cc  in  (37) 
will  always  be  correct. 

The  methods  of  deriving  c  from  the  time  observations  will 
be  treated  later  (§§  100-112). 

Correction  for  Rate. 

99.  If  the  rate  of  the  chronometer  is  known  to  be  large,  it 
may  be  necessary  to  apply  a  correction  to  each  observed  time 
to  reduce  it  to  the  mean  epoch  of  the  set.  The  correction 
required  is  the  change  in  the  error  of  the  chronometer  in  the 
interval  between  the  observation  and  the  mean  epoch  of  the 
set.  If  the  rate  of  the  chronometer  is  less  than  Is  per  day 
and  the  interval  in  question  is  not  more  than  3Om,  the  greatest 
correction  for  rate  will  be  os.O2.  In  such  a  case,  if  the  cor- 
rection for  rate  is  ignored,  the  computed  correction  to  the 

understanding  of  them)  as  will  serve  him  most  efficiently  in  making  cer- 
tain astronomical  determinations  with  portable  instruments.  The  formulae 
furnished  are  sufficiently  accurate  for  his  purpose.  The  degree  of  accu- 
racy is  roughly  indicated  to  give  him  a  basis  for  confidence.  The  alterna- 
tive procedure  is  to  derive  the  exact  formulae,  at  a  large  expenditure  of 
time  and  mental  energy;  to  find  that  said  formulae  are  too  complicated  for 
actual  use  in  computation;  to  simplify  them  by  dropping  terms  and  making 
transformations  that  are  approximate;  and  to  arrive  finally,  when  ready 
for  actual  numerical  computation,  at  the  same  simple  formula  as  are  here 
derived  directly  (or  their  equivalents  in  simplicity  and  inaccuracy).  This 
procedure  furnishes  more  mathematical  training  than  that  adopted  in  the 
text.  But  mathematical  training  is  not  the  primary  object  of  this  treatise. 


I2O  GEODETIC  ASTRONOMY.  §  IOI. 

chronometer  will  be  sensibly  exact,  but  the  computed  prob- 
able errors  will  be  slightly  too  large. 

If  the  rate  of  the  chronometer  is  very  large,  it  may  even 
be  necessary  to  apply  a  rate  correction  to  the  reduction  to  the 
mean  line,  in  case  of  an  incomplete  transit,  as  derived  in  §  92. 

Computation  of  Azimuth,  Coll imat ion,  and  Chronometer  Correc- 
tions, without  the  Use  of  Least  Squares. 

100.  Having  corrected  each  observed  time  of  transit  for 
inclination  and  aberration,  the  azimuth  error  a  and  the  colli- 
mation  error  c,  as  well  as  the  required  chronometer  correction, 
may  be  derived  from  the  observations  by  writing  an  observa- 
tion equation  of  the  following  form  for  each  star  observed, 

ATc  +  aA  +  cC-  (a  -  37)  =  o,  .      .      .     (38) 

forming  the  corresponding  normal  equations,  and  solving  for 
the  required  quantities,  ATC  the  chronometer  correction,  a, 
and  c.  In  (38)  a  is  the  apparent  right  ascension  of  the  star 
(reduced  to  mean  time  if  a  mean-time  chronometer  is  used), 
and  Tef  is  the  observed  chronometer  time  of  transit  of  the 
star  corrected  for  diurnal  aberration,  inclination ,  and  rate  of 
chronometer. 

This  least  square  process  is  rather  laborious,  and  a  shorter 
method  is  desirable  for  obtaining  approximate  results.  Such 
a  short  method  *  without  least  squares  will  now  be  treated. 
It  is  a  method  of  successive  approximations  to  the  required 
results. 

101.  The  exact  form  of  the  computation  is  shown  below 
in  a  numerical  example  dealing  with  the  observations  shown 
in  §  9'- 

*  This  method,  which  has  been  in  continual  use  in  the  field  on  the 
longitude  parties  of  the  Coast  and  Geodetic  Survey  for  many  years,  was 
devised  in  the  '7o's  by  Mr.  Edwin  Smith,  then  an  aid  on  that  Survey. 


§  101.      COMPUTATION    WITHOUT  LEAST  SQUARES.  121 


STATION  :  Washington,  D.  C.  DATE  :  May  17,  180,6. 

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122  GEODETIC  ASTRONOMY.  §  104. 

102.  The  first  five  columns  of  the  main  portion  of  the 
computation  are  compiled  from  §  91,  and  from  the  table  of 
§  299  (factors  Ay  B,  C).     The  remaining  columns  are  filled 
out  after  the  computation  of  a  and  c,  shown  in  the  lower  part 
of  the  tabular  form,  is  completed. 

103.  It  should  be  noted  that  the  five  stars  of  each  group, 
observed   in  one  position  of  the  instrument,   have  been   so 
selected  that  one  is  a  slowly-moving  northern  star  at  a  con- 
siderable distance  from  the  zenith;  while  the  other  four  are 
all  comparatively   near  the    zenith,    some   transiting  to   the 
northward  of  it  and  some  to  the  southward,  and  so  placed 
that  their  mean  azimuth  factor  (A)  is  nearly  zero.     These 
four  stars  of  each  group  are  for  convenience  called  time  stars, 
since  the  determination  of  the  time  falls  mainly  upon  themr 
while  the  slowly-moving  star  serves  to  determine  the  azimuth 
error  of  the  instrument  and  is  called  the  azimuth  star. 

104.  In  the  computation  *  to  derive  c  and  a,  the  four  time 
stars  in  each  position  of  the  instrument  are  combined  and  treated 
as  one  star,  by  taking  the  means  of  their  (a  —  77)'s  and  of 
their  factors  C  and  A,  respectively,  the  means  being  written 
below  the  separate  stars  in  the  computation  form,  together 
with  the  azimuth  starsj   On  the  assumption  that  the  means 
of  the  time  stars  in  the  two  positions  of  the  instrument  are 
equally  affected  by  the  azimuth  error,  the  first  approximation 
to  c  is  found  by  dividing  the  difference  between  the  two  mean 
values  of  a  —  Tc'  by  the  difference  between  the  two  mean 
<7s.     Or 

(<*  —  Tc'}w—  (a  —  TC)E 


W 


(39) 


*This  example  of  the  method  of  computing  a  and  c  without  least 
squares,  and  much  of  the  explanation  of  it,  is  taken  with  little  modifica- 
tion from  Appendix  No.  9  of  the  Coast  and  Geodetic  Survey  Report  for 
1896,  by  Asst.  G.  R.  Putnam, 


§105-      COMPUTATION   WITHOUT  LEAST  SQUARES.  12$ 

In  the  example  in  hand 

-  3-94-  (-4-07)       +0.13 


Using  this  approximation  to  c,  the  correction  Cc  is  then 
subtracted  from  the  a  —  Tc'  of  the  means  of  the  time  stars 
and  of  the  azimuth  stars,  and  the  values  of  a  —  27  —  Cc 
obtained. 

Separate  values  for  the  azimuth  error  of  the  instrument 
are  then  derived  for  each  position  of  the  instrument  as  fol- 
lows, upon  the  assumption  that  the  difference  between  the 
(a  —  TJ  —  Cc)  for  the  mean  of  the  time  stars  and  for  the 
azimuth  star  of  a  group  is  due  entirely  to  azimuth  error. 
Upon  this  assumption,  for  each  position  of  the  instrument 

\a  —    *<           £-^)time  stars (**          *«            ^jazimuth  stars 
*=  —  — — . 

•"time  stars          -^azimuth  stars 


and 


Numerically,  in  the  present  case 

_  —  4.00  —  (—  4.64)  _  +  0.64  _ 
"  _|_  0.08  —  (—  1.03)  ~~  -f-  1. 1 1 


_4.oo_(_  5.23)  _  + 1.23  _ 

--- -         -  +  °  '486. 


o.oo—  (—  2.53)        +2.53 


(42) 


With  these  approximate  values  of  aw  and  aB  the  correc- 
tions Aa  are  applied,  giving  the  values  a  —  Tcf  —  Cc  —  Aa 
in  the  last  column  in  the  lower  part  of  the  computation  form. 

105.  If  these  do  not  agree  for  the  two  positions  of  the 
instrument,  it  indicates  that  the  mean  values  of  a  —  Tcf  used 
in  (40)  in  deriving  c  were  not  equally  affected  by  the  azimuth 
error,  so  that  their  difference  was  not  entirely  due  to  c,  as  was 
assumed  in  using  (39).  A  second  approximation  to  the  true 
value  of  c  may  now  be  obtained  by  considering  the  differences 


124  GEODETIC  ASTRONOMY.  §  10$. 

in  the  last  column  to  be  due  to  error  in  the  first  approximate 
value  of  c\  substituting  from  that  column  in  formula  (39);  and 
thus  obtaining  a  correction  to  the  first  approximate  c.  Thus 
in  the  present  case  the  second  member  of  (39)  becomes 

—  4.05  —  (—  4.00)       —  0.05 

+  1.25  -(-  1.32)  =+^7=        °°-OI9-      '     («) 

The  second  approximation  to  the  true  value  of  c  is  then 
-f-  0s. 05 1  —  os. 019  =  +  o8. 032. 

Proceeding  as  before,  improved  values  for  aw  and  aB  are 
found  by  the  use  of  formula  (41).  Thus  in  the  present  case 
there  are  obtained  as  second  approximations 


and 


—  3.98  —  (—  4.60)       +  0.62 
+0.o8- (-i.-^)  =  +TTl 

-4.03-(-  5.31)       +1.28 


(44) 


This  process  of  making  successive  approximations  to  the 
values  of  c  and  a  may  be  continued  until  the  values  (a  —  Tc' 
—  Cc  —  Ad)  show  a  sufficiently  good  agreement^Th  general, 
with  a  well-chosen  time  set,  the  final  value  for  ATC  will  not  be 
changed  by  as  much  as  O8.oi  by  any  number  of  approxima- 
tions made  after  the  above  agreement  has  been  brought  within 
the  limit  os.O5. 

When  satisfactory  values  for  c,  aw,  and  aE  have  been 
obtained,  the  corrections  Cc  and  Aa  are  applied  separately  to 
each  star,  as  shown  in  the  sixth,  seventh,  and  eighth  columns 
of  the  upper  part  of  the  computation  form,  and  the  values  of 
the  chronometer  correction  ( A  Tc)  derived  separately  from  each 
star.  The  residuals  furnish  a  check  on  the  computation. 


§  IO6.      COMPUTATION   WITHOUT  LEAST  SQUARES.  12$ 

Any  large  error  in  observation  or  computation  will  be  indi- 
cated by  the  residuals,  and  may  often  be  located  by  a  careful 
study  of  them.  The  mean  value  of  ATC  is  the  required 
chronometer  correction  at  the  epoch  of  the  mean  of  the 
observed  chronometer  times. 

106.  A  study  of  the  above  process  of  successive  approxi- 
mation to  the  values  of  c,  aw,  and  aB  shows  that  the  rapidity 
with  which  the  true  values  are  approached  depends  upon 
three  .conditions.  The  mean  A  for  the  time  stars  for  each 
position  of  the  instrument  should  be  as  nearly  zero  as  possi- 
ble. In  each  position  of  the  instrument  the  A  for  the  azimuth 
star  should  differ  as  much  as  possible  from  the  A  for  the  mean 
time  star,  while  corresponding  C's  should  differ  as  little  as 
possible.  The  last  two  conditions'  are  difficult  to  satisfy 
simultaneously,  but  the  fact  that  both  must  be  considered 
leads  one  to  avoid  observing  sub-polars.  The  conditions  here 
stated  show  why  the  stars  for  the  above  time  set  were  chosen 
as  indicated  in  §  103.  It  is  not  advisable  to  spend  time  in 
observing  more  than  one  azimuth  star  in  each  half  set. 

It  should  be  noted  that  the  choice  of  stars  indicated  above 
also  insures  the  maximum  degree  of  accuracy  in  the  determi- 
nation of  A,TC  for  a  given  expenditure  of  time,  regardless  of 
the  method  of  computation. 

The  two  things  which  especially  commend  this  approxi- 
mate method  of  computing  time  to  those  observers  who  have 
used  it  much  in  the  field  are  the  rapidity  with  which  the 
computation  may  be  made  (especially  when  Crelle's  multipli- 
cation-tables are  used),  and  the  accuracy  which  results  from 
the  fact  that  the  derived  values  o.f  a  and  c  depend  upon  all 
the  observations,  and  not  upon  observations  upon  a  few  stars 
only,  as  is  frequently  the  case  with  other  approximate 
metnods. 


126  GEODETIC  ASTRONOMY.  §  107. 

Computation  of  the  Azimuth,  Collimation,  and  Chronometer 
Corrections  by  Least  Squares. 

107.  We  start  with  the  observation  equations  *  indicated 
in  (38).  For  lamp  west  each  of  these  equations  are  of  the 
form 


Cc-(a-  7*/)  =  o,     .      .     (45) 
and  for  lamp  east,  of  the  form 

ATC  +  AEaE  +  Cc  -  (a  -  Tcf)    =  o.     .      .     (46) 

The  subscripts  added  to  A  discriminate  between  factors 
applying  to  stars  observed  with  lamp  east  and  those  observed 
with  lamp  west.  This  is  done  for  the  purpose  of  avoiding 
confusion  in  the  normal  equations.  The  parenthesis  (a  —  Tc') 
is  an  approximate  value  for  the  clock  correction  after  taking 
account  of  inclination,  rate,  and  aberration.  It  is  the  observed 
quantity.  The  coefficients  A  and  C  may  be  obtained  from 
the  table  in  §  299. 

Treating  the  observation  equations  all  together  as  a  single 
group,  as  many  equations  as  stars,  the  four  derived  normal 
equations  are  of  the  form 

+  2Cf        -  2(a  -  Tc')      =  o; 

+  2AwCc-2Aw(a  -7y)  =  o;. 

Tc  +  2A*£aE  +  2AECc  -  2A£(a  -  T/)  =  o, 

c  +  2CA  waw  +  ^CAEaE  +  2C*c  -2C(a-  Tc')  =  o. 

The  solution  of  these  equations  gives  the  required  quan- 
tities ATe,  aw,  aE,  and  c. 

To  obtain  the  probable  error  of  a  single  observation  sub- 

*  Observation  equations  are  also  called  conditional  equations  by  some 
authors. 


§  108.  LEAST  SQUARE   COMPUTATION.  I2/ 

stitute  these  values  back  in  the  observation  equations,  (45) 
and  (46),  and  obtain  residuals  vlt  vv  v^  .  .  .  ,  one  for  each 
equation.  The  probable  error  of  a  single  observation  is 


=  0.674^7- — _,    ....  (48) 


in  which  n0  is  the  number  of  observations,  and  nv  is  the  num- 
ber of  normal  equations  (and  of  unknowns). 

To  obtain  the  probable  error,  e0,  of  the  computed  dTc, 
proceed  as  follows :  Rewrite  the  normal  equations  (47),  putting 
Q  in  the  place  of  ATC,  —  i  in  the  place  of  —  2(a  —  Te'), 
and  o  in  the  place  of  the  other  absolute  terms.  Solve  the 
resulting  equations  for  Q.  Notice  that  since  all  the  coeffi- 
cients in  these  new  equations  are  just  as  before,  it  is  only  that 
part  of  the  computation  which  deals  with  the  absolute  terms 
that  is  changed  in  the  solution  of  the  normal  equations. 

e.  =  eVQ. (49) 

The  form  of  the  normal  equations,  and  method  of  com- 
puting the  probable  error,  are  here  stated  for  convenience  of 
reference.  For  the  corresponding  reasoning  the  student  must 
depend  upon  his  knowledge  of  least  squares,  the  methods  here 
given  being  the  ordinary  least  square  methods  for  dealing 
with  a  set  of  observation  equations  in  case  there  are  no  rigid 
conditions  to  be  satisfied.* 

108.  As  a  concrete  illustration  of  this  least  square  adjust- 
ment for  determining  ATe  we  may  take  the  set  of  observations 
given  in  §  91,  from  which  ATC  has  been  computed  without 

*See  Wright's  Adjustment  of  Observations  (Van  Nostrand,  New  York), 
or  Merriman's  Least  Squares  (John  Wiley  &  Sons,  New  York). 


128  GEODETIC  ASTRONOMY.  §  IO8. 

the  use  of  least  squares  in  §  106.     The  observation  equations 
are 


V 

V* 

ATC  +  Q.02aHr                  + 

.26; 

-f-  o8.  07  =  o 

—  OMo 

0.0100 

ATC  -  0.300^                   + 

.56^ 

-|-  o  .09  =  o 

4~  °  -05 

0.0025 

4Te  +  o.36a,y                   + 

.06^ 

—  0.31=0 

-f-  o  .09 

0.0081 

ATC  +  o.22a                       + 

.13^ 

—  0  .11   =  0 

—  o  .03 

0.0009 

ATc-i^aw                   + 

.36, 

+  o  .52  =  o 

0  .00 

0.0000 

^7V                     +0.250^  — 

.!!<: 

—  o  .06  =  o 

—  0  .01 

O.OOOI 

^T;              +0.35^- 

.06^ 

—  o  .19  =  o 

H~  °  -°7 

0.0049 

Z/7V                       —  o.20aE  — 

.46. 

+  o  .23  =  o 

—  o  .06 

0.0036 

ATC                     —o.3SaE- 

.64* 

-f-  o  .29  =  o 

—  0  .02 

0.0004 

ATc                    —  2.53^£  —  L 

j.i8r 

-f  i  .44  =  o 

-f-  o  .01 

O.OOOI 

Sum  =  0.0306  =  2v* 

From  the  absolute  term  in  each  equation  4s. oo  has  been 
dropped,  as  is  frequently  the  case  in  least  square  computa- 
tions, for  the  purpose  of  shortening  the  numerical  work.  The 
true  value  of  —  (a  —  Te')  is  then,  in  each  case,  that  written 
above,  -f-  4s.oo. 

The  four  normal  equations  formed  from  the  above  obser- 
vation equations  in  the  usual  way  are 

+  iQ.oojT].  —  0.73^^  —    2.$iaB  —    2.08^+1.97  =  0; 
0.73^7;+  i. 33*^  —    2.24^—0.70  =  0; 

—  2.51/77;  +    6.77^+ 10. 84<;  —  3.88  =  o; 

—  2.08J7;  —  2.24^^+  10.84^  +  36.64;:  —  5.56  =  o. 

The  solution  of  these  equations  for  the  unknowns  gives 
aw=  +  os. 568,  0*  = +  o".5ii,  c=+os.034,  and  4TC  = 
—  08.020,  which  combined  with  the  4s. oo  which  was  dropped 
to  ease  the  numerical  work  gives  ATC  =  —  4s. 020. 

If  these  values  are  now  substituted  in  the  observation 
equations,  §  108,  the  residuals  (v)  there  shown  are  obtained. 


§  HO.  LEAST  SQUARE   COMPUTATION.  12$ 

From  these  the  probable  error  of  a  single  observation,  see 
formula  (48),  is 


The  modified  normal  equations  being  solved  for  Q  as  indi- 
cated in  §  107,  its  value  is  found  to  be  0.1158. 

Hence  the  probable  error  of  the  result  (4TC)  is,  see 
formula  (49), 

e0  =  ±  0.048  1/0.1158  =  ±  os.oi6. 

109.  If,   instead  of  computing  a    separate  value    for  the 
azimuth  error,  a,  for  each  of  the  positions  of  the  telescope 
axis,  before  and  after  reversal,  the  azimuth  error  is  assumed 
to  be  the   same   throughout   the   whole  set,    the   principles 
involved  in  the  computation  are  the  same  as  before;  the  dis- 
tinction between  aw  and  aE  is  dropped;   there  are  but  three 
unknowns  and  three  normal  equations  instead  of  four;  and 
the  work  of  solving  the  normal  equations  is  correspondingly 
shortened.     The    loss   of    accuracy   in   the   computed    result 
depends   upon  the  magnitude  of  the  actual   change   in  the 
azimuth    error   at  reversal.      If   no   more   than  six  stars  are 
observed  in  a  set,  it  may  be  advisable  to  use  this  process  so 
as  to  reduce  the  number  of  unknowns. 

110.  Experience  shows  that  the  process  outlined  in  §§  105— 
106  gives  such  an  accurate  value  for  c  that  the  value  subse- 
quently derived  from  a  least  square  adjustment  is  found  to  be 
substantially  identical  with  it.    When  such  a  preliminary  com- 
putationhasbeen made,theleast  squareadjustmentisshortened 
considerably,  with  little  loss  of  accuracy,   by  accepting  this 
preliminary  value  of  r,  applying  the  collimation  corrections 
(as  well  as  the  inclination,  rate,  and  aberration  corrections) 
before  the  least  square  adjustment,    and   treating  the  clock 


GEODETIC  ASTRONOMY.  §  I IO. 

correction  and  the  two  azimuth  errors  as  the  only  unknowns. 
It  is  well  in  this  case  to  treat  each  half  set  separately.  The 
discrepancy  between  the  two  values  for  the  clock  correction 
thus  derived,  when  reduced  for  clock  rate  to  the  same  epoch, 
indicates  the  amount  of  error  in  the  assumed  value  for  c. 

To  illustrate  this  method  we  may  use  the  same  set  of 
observations  as  in  §§  91,  101.  Let  it  be  assumed  that  the 
preliminary  computation  shown  in  §  101  has  been  made,  and 
let  the  value  -f-  os.O32  for  c  given  there  be  accepted  as  a  basis 
for  this  computation.  The  observation  equations  now  become 


-  0.02a^  +  OMI  =  O 
ATC  —  0.30^  +  o  .14  =  o 

ATC  +  0.360^—  o  .28  =  o  \  For  the  first  half  of  set. 

ATc-\-  o.22aw  —  o  .07  =  o 
A  Tc  —  1.030      +  o  '6°  =  ° 


—  o  .10  =  o 

ATC  +  o.35a£    —  o  .22  =  o 

ATC  —  o.2oa£  -f-  o  .18  =  o  }•  For  the  second  half  of  set. 

A  Tc  —  0.380^   -f  o  .24  =  o 
ATC  —  2.$3aE   -f  i  .31  =  o 


The  normal  equations  for  the  first  half  of  the  set  are 


—  0.730^  +  0.50  =  o; 
—  0.73.47;+  1.330^—  0.75  =  o; 


and  for  the  second  half  of  the  set 


5.00J7;  —  2.51^  +  1.41  =o; 
6.770*  —  3-54  =  O. 


The  solution  gives  for  the  first  half-set  ATC=  —  os.oi9, 
aw  =  +  o8.  5  5  3,  and  Q  —  0.217,  and  for  the  second  half-set 
ATC  =  —  os.  024,  aR  =  +  o8.  5  14,  and  Q  =  0.246. 

The  probable  error  of  a  single  observation  derived  from 


§  III.  UNEQUAL    WEIGHTS.  131 

the  first  half  set  is  ±  os.O58,  and  from  the  second  ±  os.O32. 
The  probable  error  of  ATC  from  the  first  half  set  is  ±  os.O27, 
and  from  the  second  ±  O8.oi6.  The  final  result  from  the 
complete  set  is,  by  this  method  of  computation,  ATC  = 

—  4S.022  ±  08.0l6. 

The  difference  between  the  two  values  for  ATC  derived 
from  the  two  halves  of  the  set  serves  to  indicate  the  degree 
of  accuracy  of  the  assumed  value  of  c. 

Introduction  of  Unequal  Weights. 

111.  In  the  preceding  treatment  it  has  been  tacitly 
assumed  that  all  observations  are  of  equal  weight.  But 
incomplete  transits  should  be  given  less  weight  than  complete 
transits.  For  if  only  a  few  lines  of  the  reticle  are  observed 
upon,  evidently  the  accidental  errors  made  in  estimating  the 
times  of  transit  across  the  separate  lines  will  not  be  eliminated 
to  as  great  an  extent  as  if  all  the  lines  were  observed.  Then, 
too,  the  image  of  a  star  of  large  declination  moves  much  more 
slowly  across  the  reticle  than  does  the  image  of  an  equatorial 
star,  and  it  is  therefore  more  difficult  to  estimate  the  exact 
time  of  its  transit  across  each  line.  If  it  is  found  by  the 
investigation  of  many  records  for  such  slow-moving  stars  that 
the  error  of  observation  is  larger  for  such  stars  than  for 
equatorial  stars,  it  is  proper  to  give  them  less  weight  in  the 
computation  of  time.  An  extended  discussion  of  this  matter 
of  weights  may  be  found  in  the  Annual  Report  of  the  Coast 
and  Geodetic  Survey  for  1880,  pp.  213,  235-237.  It  suffices 
for  our  purpose  here  to  give,  in  slightly  abridged  form,  the 
tables  of  relative  weights  which  were  derived  from  that  dis- 
cussion (see  §§  302,  303).  The  weights  as  given  in  these 
tables  were  deduced  for  transit  instruments  having  a  clear 
aperture  of  object-glass  from  if  to  2f  inches,  and  a  magnifying 
power  from  70  to  100  diameters.  It  may  be  extended  with 


132  GEODETIC  ASTRONOMY.  §  112. 

little  error  to  instruments  of  the  same  nature  which  are  con- 
siderably larger  or  smaller. 

112.  To  introduce  the  unequal  relative  weights,  wt  into 
the  least-square  adjustment,  it  is  necessary  to  multiply  each 
observation  equation  by  Vw,  and  to  make  the  usual  subse- 
quent modifications  in  the  least  square  computation.  These 
modifications  are  indicated  in  the  following  example,  —  the 
same  problem  as  that  treated  in  §  108,  without  the  use  of 
unequal  weights.  If  an  incomplete  observation  is  made  upon 
a  slow  star,  so  that  both  the  tables  of  §  302  and  of  §  303  must 
be  used,  first  multiply  the  two  relative  weights  w  together, 
and  then  take  the  square  root  of  that  product  as  the  multi- 
plier for  the  observation  equations. 

The  square  roots  of  the  weights  given  to  the  ten  stars,  in 
order  of  observation,  are  respectively  0.9,  0.8,  i.o,  0.9,  0.6, 
0.9,  i.o,  0.8,  0.8,  and  0.3.  The  observation  equations  shown 
in  §  108  are  multiplied  by  these  factors  respectively.  The 
normal  equations  resulting  from  the  weighted  observation 
equations  so  obtained  are 


-\-  0.53^  +  os.  14  =  o; 

—  0.57^:  —  os.34  =  o; 

—  0.04/47;  +  o.S?aE  +  0.93*:  —  0s.  50  =  o; 

+  0.53^7;- 


The  solution  of  these  equations  gives  ATC=  —  o*.oi9> 
aw  =  +  0s.  583,  aE  =  +  0s.  544,  and  c  =  +  os.O33. 

The  probable  error  of  an  observation  of  weight  unity  is 


/  ^W  V  /O.O223 

0.674.  A'/'-         -—0.674  A/ -==:fco*.O4i. 

\   n0  —  nv  *  V    10  — 4 


The  probable  error  of 


ATC  =  e0  =  e  VQ  =  ±  os.04i  Vo.i47  =  ±  os.oi6. 


§114-  AUXILIARY  OBSERVATIONS.  133 

113.  Unless  an  extreme  degree  of  accuracy  is  required,  the 
assumption  that  all  observations  are  of  equal  weight  is  suffi- 
ciently exact.      The  introduction  of  unequal  weights  adds  so 
little  to  the  accuracy  of  the  computation  that  economic  con- 
siderations will  often  indicate  that  the  least  square  adjustment 
should  be  made  on  the  basis  of  equal  weights.* 

Auxiliary  Observations. 

114.  Aside  from  the  observations  and  computations  which 
have  been  treated  in  detail,  certain  others  are  necessary  for 
the  determination  of  the  instrumental  constants  which  have 
been  assumed  in  the  preceding  treatment  to  be  known. 

The  equatorial  intervals  of  the  lines  of  the  reticle  may  be 
determined  from  any  series  of  complete  transits,  i.e.,  observa- 
tions in  which  the  transit  of  each  star  was  observed  across 
every  line.%  A  special  series  of  observations  is  not  required, 
for  the  complete  transits  of  the  particular  series  of  time 
observations  under  treatment  may  be  utilized  for  this  purpose 
in  addition  to  using  them  to  determine  the  clock  correction. 
For  every  complete  transit  every  term  in  equation  (20)  (see 
§  92),  namely,  in  —  (tn  —  tm)  cos  #,  is  known  except  in.  Every 

*  The  computation  of  a  series  of  time  observations  taken  by  the  author 
on  the  shore  of  Chilkat  Inlet,  Alaska  (in  latitude  59°  10'),  in  1894,  was  made 
in  the  field  by  least  squares,  giving  all  stars  equal  weight,  regardless  of 
their  declinations  and  of  the  number  of  missed  lines.  In  the  final  com- 
putation subsequently  made  at  the  Coast  and  Geodetic  Survey  Office  in 
Washington  unequal  weights  were  assigned.  In  the  series  there  were  46 
sets,  each  consisting,  generally  speaking,  of  observations  upon  10  stars. 
The  average  difference,  without  regard  to  sign,  between  the  chronometer 
corrections  as  computed  in  the  two  ways  from  the  same  set  of  observations 
was  o».04.  This  is  about  equal  to  the  probable  error  of  the  clock  correc- 
tion computed  from  a  set.  But  it  must  be  remembered  that  the  conditions 
were  extreme.  On  account  of  the  high  latitude  of  the  station  many  of  the 
stars  were  slow-moving  stars  (even  those  observed  in  the  zenith).  There 
was  so  much  interference  by  clouds  that  complete  observations  on  all  the 
stars  were  secured  on  only  10  nights  out  of  the  46,  and  observations  on  a 
single  line  only  of  the  reticle  were  not  infrequent. 


134 


GEODETIC  ASTRONOMY. 


complete  transit  observed  furnishes,  then,  a  determination  of 
the  equatorial  interval  of  every  line.  The  transit  of  a  slow- 
moving  star  gives  a  more  accurate  determination  of  the 
equatorial  intervals  than  the  transit  of  a  star  of  small  declina- 
tion, for  the  errors  in  observing  tn  do  not  increase  so  rapidly, 
with  increase  of  declination,  as  cos  d  decreases.  For  this 
reason  some  observers  prefer  to  make  a  special  series  of 
observations  for  equatorial  intervals  using  stars  of  large  decli- 
nation only.  In  computing  and  using  the  equatorial  intervals 
it  must  be  borne  in  mind  that  when  the  telescope  axis  is 
reversed  in  its  Ys  the  order  in  which  the  star  transits  across 
the  lines  is  reversed,  and  also  the  algebraic  sign  of  the  equa- 
torial interval  of  each  line. 

115.  The  portion  of  a  set  of  observations  given  below  will 
serve  to  show  how  the  pivot  inequality,  pi  (see  §  94),  is 
determined  by  a  series  of  readings  of  the  striding  level,  upon 
the  telescope  axis  placed  alternately  in  each  of  its  two  possi- 
ble positions  with  clamp  west  and  clamp  east  (the  clamp 
instead  of  the  lamp  being  here  used  to  indicate  the  position 
of  the  axis). 

OBSERVATIONS  FOR  INEQUALITY  OF  PIVOTS  OF 

TRANSIT  NO.  4. 
STATION  :     Seaton,  Washington. — G   W.  D.,  observer. — June  19,  1867. 


u 

CLAMP  WEST.                             CLAMP  EAST.V 

a 

g 

Object-glass  S. 

Object-glass  N. 

*.-*« 

V 

I 

Level. 

i(2«/  -  2,e\ 

Level. 

K2?«  -  Ze) 

4 

•o 

V 

a 

W.  end. 

E.end. 

bv> 

W.end. 

E.end. 

be 

< 

H 

0 

h.  m. 

o 

d. 

d. 

d. 

d. 

d. 

d. 

d. 

33 

10.30  A.M. 

73 

60  o 

64.0 

+  0.600 

59-o 

65.2 

-0.425 

—  o.  256 

65.2 

58.8 

64.0 

59'5 

SO 

45A.M. 

72 

65.0 

59-o 

+  0.950 

64  o 

59-5 

—  0.250 

—  o  .  300 

60.8 

63.0 

59'Q 

64.5 

45 

50  A.M. 

72.5 

60.8 
66.0 

63-0 
58.0 

+  1.450 

59-5 
04.0 

64.0 
60.0 

—  0.125 

-  0.394 

40 

11.00  A.M. 

72.8 

65.0 

58.8 

+  1.050 

64.0 

60.0 

—  0.175 

-0.306 

61  .0 

63.0 

59  3 

64.0 

IS 

05  A.M. 

73 

60.5 

63.0 

-f  1.200 

59-2 

64  o 

-0-575 

-  o  444 

65-5 

58    2 

63.0 

60.5 

§  Il6.  THE   LEVEL    VALUE.  135 

The  value  of  one  division  of  this  striding  level  was  known 
to  be  I ".05.  The  whole  set,  of  which  this  is  a  part,  gave  for 
a  mean  value  of  pi  —  0.337  divisions  of  the  level  =  o//.354  = 

Os.024. 

116.  The  most  accurate  way  of  determining  the  value  of 
one  division  of  a  level  is  by  means  of  what  is  known  as  a 
level-trier  or  level-tester.  The  level-trier  is  essentially  a  bar 
supported  at  one  end  upon  two  pivots  so  that  it  is  free  to 
rotate  about  that  end  in  a  vertical  plane,  and  carried  at  the 
other  end  by  a  good  micrometer  screw  with  its  axis  vertical. 
The  length  of  the  bar  between  supports  and  the  pitch  of  the 
screw  being  known,  the  change  of  inclination  of  the  bar 
corresponding  to  one  turn  of  the  screw  is  known.  To  deter- 
mine the  value  of  a  division  of  a  level  it  is  placed  upon  the 
bar,  and  movements  of  the  bubble  corresponding  to  successive 
small  movements  of  the  micrometer  screw  are  observed. 
Both  ends  of  the  bubble  must  of  course  be  read,  as  its  length 
is  apt  to  change  rapidly  with  changes  of  temperature.  By 
comparing  successive  movements  of  the  bubble  corresponding 
to  equal  successive  movements  of  the  screw  the  uniformity  of 
the  value  of  a  level  division,  or,  in  other  words,  the  constancy 
of  the  radius  of  curvature  of  the  upper  inner  surface  of  the 
level  tube,*  may  be  inferred.  Observations  of  the  length  of 
time  required  for  the  bubble  to  come  to  rest  in  a  new  position 
also  give  an  indication  of  the  value  of  the  level  as  an  instru- 

*  The  longitudinal  section  of  the  upper  inner  surface  of  a  level  tube  is 
made  as  nearly  a  perfect  circle  as  possible.  If  the  student  will  consider 
how  great  is  this  radius  of  curvature  in  a  sensitive  striding  level,  he  will 
appreciate  to  a  certain  extent  the  wonderful  accuracy  with  which  this  sur- 
face must  be  ground.  He  will  also  understand  why  small  deformations  of 
the  level  tube  by  unequal  changes  of  temperature  have  such  a  marked 
effect  upon  the  movement  of  the  bubble.  The  radius  of  curvature  for  a 
level  of  which  each  division  is  two  millimeters  long  and  is  equivalent  to 
one  and  a  quarter  seconds  of  arc, — a  common  type  of  level, — is  more  than 
three  hundred  meters  (about  a  thousand  feet). 


136  GEODETIC  ASTRONOMY.  §H9- 

ment  of  precision.  This  time  is  greater  the  smaller  is  the 
value  of  a  division  (of  a  given  length)  expressed  in  arc,  but 
for  levels  of  the  same  division  value,  is  less  the  more  perfect  is 
the  inner  upper  surface.  If  the  level  tube  is  so  held  in  its 
metallic  mounting  that  there  is  any  possibility  that  it  may  be 
put  under  stress  by  a  change  of  temperature,  it  is  advisable  to 
determine  the  value  of  a  division  with  the  tube  in  its  mounting 
at  two  or  more  widely  different  temperatures.  It  may  be 
well  also  to  determine  whether  changing  the  length  of  the 
bubble,  by  changing  the  amount  of  liquid  in  the  chamber  at 
the  end  of  the  level  tube,  changes  the  apparent  value  of  one 
division. 

117.  If  an  observer  is  forced  to  determine  the  value  of  a 
level  division  in  the  field,  remote  from  a  level-trier,  after  some 
accident  let  us  say,  which  leads  to  replacing  an  old,  well-known 
(but  broken)  level  by  another  of  which  the  value  is  unknown, 
his  ingenuity  will  lead  him  to  devise  a  method  of  utilizing 
whatever  apparatus  is  at  his  disposal.     The  three  methods 
given  below  will  be  found  suggestive. 

118.  If  a  telescope  having  an  eyepiece  micrometer  similar 
to  that  of  a  zenith  telescope  (§  135),  measuring  altitudes  or 
zenith  distances,  is  available,  the  unknown  angular  value  of  a 
division  of  the  level  may  be  found  by  comparison  with  the 
known  angular  value  of  a  division  of  the  micrometer.      Place 
the  level  in  an  extemporized  mounting  fixed  to  the  telescope. 
Point  with  the  micrometer  upon   some  distant  well-defined 
fixed  object  and  read  the  micrometer  and  level.     Change  the 
micrometer  reading  by  an  integral  number  of  divisions,  point 
to  the  same  object  again  by  a  movement  of  the  telescope  as 
a  whole,    and   note   the   new   reading  of  the   level.      Every 
repetition  of  this  routine  gives  a  determination  of  the  value 
of  a  level  division. 

119.  If  in  his  instrumental  outfit  he   has  another  well- 


§  120.  THE  LEVEL    VALUE.  137 

determined  level  of  sufficient  sensibility  the  observer  may  use 
it  as  a  standard  with  which  to  compare  the  unknown  level. 
Put  the  unknown  level  in  an  extemporized  mounting  fastened 
to  that  of  the  known  level.  Adjust  so  that  both  bubbles  are 
near  the  middle  at  once.  Compare  corresponding  movements 
of  the  two  bubbles  for  small  changes  of  inclination  common 
to  both  levels. 

120.  The  following  method  *  gives  fully  as  great  precision 
as  either  of  the  other  two  outlined  above,  and  is  especially 
valuable  because  the  required  means  are  apt  to  be  at  hand  in 
the  field  even  when  the  apparatus  required  for  the  other  two 
methods  is  wanting. 

For  this  method  the  only  instrument  required  is  a  theodo- 
lite, or  an  engineer's  transit,  or  any  other  instrument  having 
both  horizontal  and  vertical  circles  (not  necessarily  with  a  fine 
graduation)  and  a  good  vertical  axis.  Mount  the  level  on  the 
plate  of  the  instrument  parallel  to  the  plane  of  the  telescope 
and  adjust  it  as  if  it  were  a  plate  level.  Make  the  vertical 
axis  truly  vertical  in  the  usual  way.  Measure  the  zenith  dis- 
tance of  some  well-defined  stationary  object,  taking  readings 
with  (vertical)  circle  right  and  circle  left  to  eliminate  index 
error.  Now  incline  the  vertical  axis  directly  toward  or  from 
the  object,  from  i°  to  3°,  by  use  of  the  foot-screws.  The 
direction  of  this  inclination  may  be  assured  by  use  of  the 
plate  level  which  is  at  right  angles  to  the  plane  of  the  tele- 
scope. Measure  the  apparent  zenith  distance  of  the  object 
again.  The  apparent  change  in  the  zenith  distance  is  evi- 
dently the  inclination  of  the  axis  to  the  vertical,  which  we 
will  call  y.  If,  now,  the  instrument  is  revolved  completely 

*  Described  in  full  by  Prof.  G.  C.  Comstock  in  the  Bulletin  of  the  Uni- 
versity of  Wisconsin,  Science  Series,  vol.  I,  No.  3,  pp.  68~74;  and  said  by 
him  to  be  due  originally  to  Braun.  Those  desiring  further  details  are 
referred  to  that  article,  from  which  this  statement  is  condensed. 


138  GEODETIC  ASTRONOMY.  §  121. 

around  its  vertical  axis,  two  positions  will  be  found  at  which 
the  bubble  of  the  level  is  in  the  middle  of  the  tube.  For 
positions  near  these  two  the  bubble  is  within  such  limits  that 
it  may  be  read.  It  is  from  readings  of  the  bubble  in  such 
positions,  in  connection  with  readings  of  the  horizontal  circle 
and  the  above  outlined  determination  of  y,  that  the  value  of 
a  division  of  the  level  is  derived. 

121.  Let  Fig.  19  "  represent  a  portion  of  the  celestial 
sphere  adjacent  to  the  zenith,  Z,  and  let  V  and  S  be  the 
points  in  which  the  axis  of  the  theodolite,  and  the  line  drawn 
from  the  center  of  curvature  of  the  level  tube  through  the 
middle  of  the  bubble,  respectively,  intersect  the  sphere." 
14  Since  the  bubble  always  stands  at  the  highest  part  of  the 
tube,  its  position,  5,  and  the  corresponding  value  of  q  are 
found  by  letting  fall  a  perpendicular  from  the  zenith  upon  the 
arc  VS,  and  in  the  right-angled  spherical  triangle  thus  formed 
we  have  the  relation  " 

tan  q  =  tan  y  cos  ft.    .      .      .      <     .      (50) 

"  Since  the  level  tube  turns  with  the  theodolite  when  the 
latter  is  revolved  in  azimuth,  while  the  positions  of  the  points 
Fand  Z  remain  unchanged,  it  appears  that  the  angle  ft  must 
vary  directly  with  the  readings  of  the  azimuth  circle."  "  If 
we  represent  by  AQ  the  reading  of  the  circle  when  the  arc  VS 
is  made  to  coincide  with  VZ,  we  shall  have  corresponding  to 
any  other  reading  A'  " 

tan  q  =  tan  y  cos  (A0  —  A1).       .      .      .     (51) 

The  value  of  A0  may  be  obtained  by  taking  the  mean  of 
any  two  readings  of  the  circle  for  which  the  bubble  stands  at 
the  same  part  of  the  tube. 

"  If  A'  and  A"  denote  slightly  different  readings  of  the 
azimuth  circle,  b'  and  b"  the  corresponding  readings  of  the 


§  121.  THE   LEVEL    VALUE.  139 

middle  of  the  bubble  on  the  level  scale,  we  may  write  two 
equations  similar"  to  (51)?  "and  taking  their  difference 
obtain  " 

sin  (q'-q"}  A'  -  A"    .     /  A'  +  A"\ 

—^  --  H$  =  2sm-        -  smL40  —  -  -    tan  y.     (52) 

cos  q  cos  q  2  \    '  / 

lt  Since  q'  —  q"  is  the  distance  moved  over  by  the  bubble, 
we  may  write  q  —  q"  =  (b'  —  b"}d,  where  d  is  the  value  of 
a  division  of  the  level,  and  transform  "  (52)  into 


_  2  tan  yc^qsm^(Af  -  A")  sin  [A0  -  \(A'  +  A")] 
sin  i"  b'-b" 

In  this  equation  cos2  q  may  usually  be  placed  equal  to 
unity.  For  greater  accuracy  the  average  value  of  q  from 
equation  (51)  may  be  used.  AQ  may  be  determined  as  indi- 
cated just  below  equation  (51).  Only  an  approximate  value 
for  it  is  required.  All  other  quantities  in  the  second  member 
of  ($3)  are  known.  Hence  d  may  be  computed.  It  simplifies 
the  computation  to  take  the  readings  at  equidistant  points 
on  the  circle.  Sin  \(A'  —  A")  will  then  be  constant. 

It  would  seem  at  first  sight  that  a  value  of  d  derived  in 
this  way  by  use  of  vertical  and  horizontal  circles  reading  to 
half-minutes  only  (say)  must  necessarily  be  crude.  If,  how- 
ever, the  inclination  of  the  vertical  axis  is  made  3°,  y  and 
its  tangent,  and  therefore  d,  may  be  determined  within  one 
four-hundredth  part.  The  readings  of  the  horizontal  circle 
do  not  need  to  be  very  refined,  because  for  the  positions  used 
a  comparatively  large  change  in  the  circle  reading  is  necessary 
to  produce  an  appreciable  change  in  the  position  of  the 
bubble.  This  method,  then,  serves  to  determine  the  level 
value  with  an  accuracy  which  bears  little  relation  to  the  fine- 
ness of  the  circle  graduations. 


140  GEODETIC  ASTRONOMY.  §  123. 

Discussion  of  Errors. 

122.  Following  the  same  general  plan  as  in  discussing  the 
errors   of  sextant   observations,   the   external  errors,    instru- 
mental errors,  and  observer's  errors  will   be  discussed  sepa- 
rately, and  then  their  combined  effect  will  be  considered. 

The  two  principal  external  errors  are  the  error  in  the 
assumed  right  ascension  of  the  star,  and  the  lateral  refraction 
of  the  light  from  the  star. 

If  only  such  stars  as  are  given  in  the  va-ious  national 
ephemerides  are  observed  for  time,  the  probable  errors  in  the 
right  ascensions  will  usually  be  on  an  average  ±  O8.O4  or 
±  os.O5,  and  no  appreciable  constant  errors  need  be  appre- 
hended from  this  source. 

From  considerations  which  need  not  be  stated  in  detail 
here,  one  is  led  to  the  conclusion  that  the  effect  of  lateral 
refraction  upon  transit  time  observations  must  be  quite  small 
in  comparison  with  the  other  errors ;  but  it  is  difficult  to 
estimate,  because  it  is  always  masked  by  other  errors  follow- 
ing about  the  same  law  of  distribution.  For  further  consid- 
eration of  this  matter  see  §  219. 

123.  Among  the  instrumental  errors  may  be  mentioned 
those  arising  from  change  in  azimuth,  collimation,  and  inclina- 
tion, from  non-verticality  of  the  lines  of  the  reticle,  from  poor 
focusing  and  poor  centering  of  the  eyepiece,  from  irregularity 
of  pivots,  and  from  variations  in  the  clock  rate. 

The  errors  of  azimuth  and  collimation  being  determined 
from  the  observations  themselves  are  quite  thoroughly  can- 
celled out  from  the  final  result,  provided  they  remain  constant 
during  the  period  over  which  the  observations  extend,  and 
provided  also  that  the  stars  observed  are  so  distributed  in 
declination  as  to  furnish  a  good  determination  of  these  con- 
stants. Their  changes,  however,  during  that  interval,  arising 


§123.  ERRORS.  141 

from  changes  of  temperature,  shocks  to  the  instrument,  or 
other  causes,  produce  errors  in  the  final  result.  It  is  in  this 
connection  that  the  stability  of  the  pier  is  of  especial  impor- 
tance. Such  changes  will  evidently  be  smaller  the  more 
rapidly  the  observations  are  made  and  the  more  carefully  the 
instrument  is  handled.  In  general  they  are  probably  small 
but  not  inappreciable. 

To  a  considerable  extent  the  same  remarks  also  apply  to 
the  inclination  error.  The  changes  in  inclination  during  each 
half-set  evidently  produce  errors  directly.  Hence  again  the 
desirability  of  rapid  manipulation.  But  the  mean  value  of 
the  inclination  is  determined  from  readings  of  the  striding 
level,  not  from  the  time  observations,  and  the  level  may  give 
an  erroneous  determination  of  the  mean  inclination.  Differ- 
ent observers  seem  to  differ  radically  as  to  the  probable  mag- 
nitude of  errors  from  this  source,  but  the  best  observers  are 
prone  to  use  the  striding  level  with  great  care.  However 
small  this  error  may  be  under  the  best  conditions  and  most 
skilful  manipulation,  there  can  be  no  doubt  that  careless 
handling  and  slow  reading*  of  the  striding  level,  or  a  little 
heedlessness  about  bringing  a  warm  reading  lamp  too  near  to 
it,  may  easily  make  this  error  one  of  the  largest  affecting  the 
result.  An  error  of  0.0002  inch  in  the  determination  of  the 
difference  of  elevation  of  the  two  pivots  of  such  an  instrument 
as  that  described  in  §  83  produces  an  error  of  o8. 1  or  more  in 
the  deduced  time  of  transit  of  a  zenith  star. 

If  the  lines  of  the  reticle  are  not  carefully  adjusted  so  as 
to  define  vertical  planes  (§  85),  stars  will  be  observed  too 
early  or  too  late  if  observed  above  or  below  the  middle  of  the 
reticle.  Such  errors  may  be  made  very  small  by  careful 

*  It  is  here  assumed  that  before  attempting  to  read  the  level  it  has  been 
in  position  long  enough  for  the  bubble  to  come  to  rest  in  the  position  of 
equilibrium. 


142  GEODETIC  ASTRONOMY.  §  124. 

adjustment  and  by  always  observing  within  the  narrow  limits 
given  by  the  two  horizontal  lines  of  the  reticle. 

Poor  focusing  of  either  the  object-glass  or  the  eyepiece 
leads  to  increased  accidental  errors  because  of  poor  definition 
of  the  star  image.  But  poor  focusing  of  the  object-glass  is 
especially  objectionable,  because  it  puts  the  reticle  and  the 
star  image  in  different  planes,  and  so  produces  parallax.  The 
parallax  error  may  largely  be  avoided  by  centering  the  eyepiece 
each  time  over  the  line  of  the  reticle  upon  which  the  star  is 
next  to  be  observed.  This  repeated  centering  should  never  be 
omitted  even  though  the  observer  may  be  confident  that  the 
focusing  is  perfect.  It  also  serves  in  a  measure  to  avoid 
errors  which  might  otherwise  be  produced  by  the  imperfec- 
tions of  the  eyepiece. 

If  the  inequality  of  the  two  pivots  has  been  carefully 
determined  as  indicated  in  §  115,  the  errors  arising  from 
defects  in  their  shapes  may  ordinarily  be  depended  upon  to 
be  negligible. 

Changes  in  the  rate  of  the  timepiece  during  a  set  of 
observations  evidently  produce  errors  in  the  deduced  clock 
correction  at  the  mean  epoch  of  the  set.  Under  ordinary 
circumstances  such  errors  must  be  exceedingly  small.  If, 
however,  an  observer  is  forced  to  use  a  very  poor  timepiece, 
or  if  clouds  interfere  so  as  to  extend  the  interval  required  for 
a  set  of  observations  over  several  hours,  this  error  may 
become  appreciable.  It  is  less  the  more  rapidly  the  obser- 
vations are  made. 

The  errors  introduced  by  irregularity  in  the  action  of  a 
chronograph  of  the  form  described  in  §  89  are  too  small  to  be 
considered,  especially  if  its  speed  is  assumed  to  be  constant 
simply  during  the  interval  between  successive  clock  breaks, 
and  the  chronograph  sheet  is  read  accordingly. 

124.  The  observer  s  errors  are  by  far  the  most  serious  in 


§  124.  ERRORS.  143 

transit  time  observations.      He  is  subject  to  both  accidental 
and  constant  errors  in  his  estimate  of  the  time  of  transit  . 

From  computations  based  upon  thousands  of  observed 
transits  it  is  known  that  an  experienced  observer  is  subject  to 
an  accidental  error  of  from  ±  os.o6  to  ±  o8.  15  in  estimating 
the  time  of  transit  of  a  star  of  declination  less  than  60°  across 
a  single  line  of  the  reticle  of  such  instruments  as  those  de- 
scribed in  §  83.  For  slower  stars  his  error,  expressed  in 
time,  is  of  course  still  greater.  If  the  observations  of  the 
transits  of  a  given  star  across  the  different  lines  of  the  reticle 
were  not  subject  to  any  error  common  to  all  the  lines,  the 
probable  error  of  the  deduced  time  of  transit  across  the  mean 
line  would  vary  inversely  as  the  square  of  the  number  of  lines 
in  the  reticle.  But  experience,  as  above,  indicates  that  there 
is  an  error  common  to  all  the  lines  of  O9.O5  to  O*.I2.  This 
error,  sometimes  called  the  culmination  error,  is  an  observer's 
error,  which  is  constant  for  the  interval  during  which  the  star 
is  transiting  across  the  reticle,  but  which  may  change  before 
the  next  star  is  observed.  Prom  the  method  by  which  this 
value  (os.O5  to  O8.I2)  was  deduced  it  also  necessarily  includes 
the  small  errors  due  to  lateral  refraction  and  irregularities  in 
clock  rate,  as  well  as  some  small  outstanding  instrumental 
errors.  The  probable  error,  r,  of  the  time  of  transit  of  a  star 
(of  declination  less  than  60°)  across  the  mean  line  of  a  reticle 
is,  therefore,  given  by  an  equation  of  the  form 

(o8.o6)a  to  (os.  1  5)' 
r9  =  (o8.  05  to  os.  12)'  +  v-  -- 


in  which  n  is  the  number  of  lines  in  the  reticle.  (Compare 
§§  in,  302,  303.)  For  a  more  extended  discussion  see 
Coast  and  Geodetic  Survey  Report  for  1880,  Appendix  No. 
14,  pp.  235,  236,  or  Doolittle's  Practical  Astronomy,  pp. 
318-322 


144  GEODETIC  ASTRONOMY.  §  I2/. 

125.  In  addition,   still,   to  these  errors  there    is  another 
which  is  constant  for  all   the  observations  of  a  set.      Every 
experienced  observer,   though  doing  his  best  to  record  the 
time  of  transit  accurately,  in  reality  forms  a  fixed  habit  of 
observing  too  late,  or  too  early,  by  a  constant  interval.      This 
interval  between  the  time  when  the  star  image  actually  tran- 
sits across  a  line  of   the   reticle  and   the   recorded   time   of 
transit  is  called  the  absolute  personal  equation  of  the  observer. 
The  difference  between  the  absolute  personal  equations  of  two 
observers  is  called  their  relative  personal  equation.     The  rela- 
tive personal  equation  of  two  experienced  observers  has  been 
known  to  be  as  great  as  Is. 2,  and  values  greater  than  os,25 
are   common.      For  a  more   detailed   discussion   of  personal 
equation,  see  §§  243,  244. 

126.  To   sum    up,   it  may  be  stated   that   the  accidental 
errors  in  the  determination  of  a  clock  correction  from  obser- 
vations with  a  portable  astronomical  transit  upon  ten  stars 
may  be  reduced  within  the  limits  indicated  by  the  probable 
error  ±  os.O2  to  ±  os.  10,  but  that  the  result  is  subject  to  a 
large  constant  error,  the  observer's  absolute  personal  equation, 
which  may  be  ten  times  as  great  as  this  probable  error. 

Miscellaneous. 

127.  In  the  field  it  is  often  necessary  to  use  other  instru- 
ments as  transits  for  the  determination  of  time.     A  theodolite 
when  so   used  is  apt  to  give  results  of  a  higher  degree  of 
accuracy  than  would  be  expected  from  an  instrument  of  its 
size  as  compared  with  the  astronomical  transits  whose  per- 
formance has  just  been  discussed, — unless,  indeed,  one  has  it 
firmly  fixed  in  mind  that  the  principal  errors  in  a  transit  time 
determination  are  those  due  directly  to  the  observer.     On  the 
other  hand,  a  zenith  telescope  of  the  common  form  in  which 
the  telescope  is  eccentric  with  respect  to  the  vertical  axis  has 


§  130.  MISCELLANEOUS.  145 

been  found  to  give  rather  disappointing  results, — perhaps 
because  of  the  asymmetry  of  the  instrument  and  of  the  fact 
that  there  can  be  no  reversal  of  the  horizontal  axis  in  its 
bearings,  but  only  of  the  instrument  as  a  whole. 

128.  The  mathematical  theory  for  the  determination  of 
time  by  the  use  of  the  transit  in  any  position  out  of  the 
meridian  has  been  thoroughly  developed.     That  practice  has 
been  advocated.     But  the  additional  difficulty  of  making  the 
computation,  over  that  for  a  transit  nearly  in  the  meridian,  and 
other  incidental  inconveniences,  much  more  than  offset  the 
fact    that    the    adjustment    for   putting    the    transit    in    the 
meridian  is  unnecessary.     The  transit  is  generally  used  in  the 
meridian  for  time,  at  least  in  this  country. 

129.  The  use  of  the  transit  for  time  in  the  vertical  plane 
passing  through  Polaris  at  the  time  of  observation  has  also 
been  advocated  and  has  been  used  to  a  considerable  extent  in 
Europe.       "  The    obvious    advantage   which    this    mode    of 
observing  possesses  lies  in  the  shorter  period  of  time  during; 
which  the  observer  depends  upon  the   stability  of  his  instru- 
mental constants.      For  meridian  observations   this  period  is 
rarely  much  less  than  half  an  hour,  while  by  the  method  sug- 
gested " — in  which  the  whole  time  set  consists  of   a  pointing; 
upon  Polaris  immediately  followed  by  an  observation  of  the. 
transit  of  a  zenith  or  southern  star  across  that   vertical  plane 
— "  it  need  never  exceed  five  minutes."*     This  method  is 
open,  to  a  less  extent,  to  the  same  objections  as  that  of  the 
preceding  paragraph.      This,  in  connection  with  the  fact  that 
it  is  rarely  used  in  this  country,  makes  its  extended  discussioa 
inadvisable  here. 

130.  If  the  transit  is  turned  at  right  angles  to  the  plane 
of   the  meridian,  in  other  words,  is  put  in  the  prime  vertical, 

*  See   Bulletin  of  the   University  of  Wisconsin,  Science  Series,  vol.  !.„, 

No.  3.  pp.  81-93. 


146  GEODETIC  ASTRONOMY.  §  133. 

an  observation  of  the  time  of  transit  of  a  star  across  the  mean 
line  of  its  reticle  furnishes  a  good  determination  of  the  lati- 
tude of  the  station  if  the  clock  correction  is  known.  Or,  if 
both  transits,  east  and  west  of  the  zenith,  are  observed,  the 
latitude  may  be  computed  without  a  knowledge  of  the  clock 
correction.  Formerly  this  method  *  was  often  used  for  the 
determination  of  latitude.  Now  it  is  almost  entirely  super- 
seded by  the  use  of  the  zenith  telescope  for  latitude. 

131.  The  Sun  or  a  planet  may  sometimes  be  observed  for 
time.      In  the  case  of  the  Sun  the  transit  of  both  the  preced- 
ing and  the  following  limb  may  be  observed,  and  the  mean 
taken  as  the  time  of  transit  of  the  center.      Both  limbs  of  a 
planet  may  possibly  be  observed  if  a  chronograph  is  used. 
Otherwise  the  preceding  and  following  limbs  may  be  observed 
alternately  on  successive  lines  of  the  reticle,  taking  care  that 
the  number  of  observations  on  each  limb  is  the  same,  and  the 
mean  of  all  taken  as  the  transit  of  the  center  across  the  mean 
line. 

132.  It  is  not  advisable  to  observe  the  Moon  for  time,  for 
its  place  is  not  well  determined.    Usually  but  one  limb  can  be 
observed,  the  other  being  either  obscure  or  invisible;  and  the 
observation  of  the  limb  on  a  side  line  of  the  reticle  is  affected 
by  the  rapid  change  in  the  Moon's  right  ascension  and  by  a 
parallax  due  to  its  comparative  nearness  to  the  Earth. 

QUESTIONS  AND   EXAMPLES. 

133.  i.  An  observer  who  is  trying  to  get  his  transit  into 
the  meridian  to   begin   observations  for  time  finds  that  an 
observation  upon  rf  Draconis  (6  =  61°  45 ')  indicates  that  his 
chronometer  is  4*. 2  fast  of  local  sidereal  time,  while  an  obser- 
vation   upon    /3    Herculis    (6  =  21°    43')    indicates    that    his 

*  For  the  detail   of  this  method  see  Doolittle's   Practical  Astronomy, 
pp.  348-377,  or  Chauvenet's  Astronomy,  vol.  u.,  pp.  238-271. 


§  133-  QUESTIONS  AND    EXAMPLES.  147 

chronometer  is  Is.  3  slow.  Assuming  that  the  instrument  is 
in  perfect  adjustment  with  respect  to  collimation  and  inclina- 
tion, how  much  must  he  turn  the  slow-motion  screw  which 
shifts  his  instrument  in  azimuth,  if  one  turn  produces  a  change 
of  200"  in  azimuth  ?  The  latitude  of  the  station  is  39°  58'. 
Is  the  object-glass  too  far  east,  or  too  far  wesc,  when  the 
telescope  is  pointing  northward  ? 

Ans.  0.37  turn.     Too  far  west. 

2.  The  star  5  Ursae  Minoris  (tf  =  76°  09')  was  observed 
to  transit  as  follows:  Line  I,  9h  48™  4is.O3:  II,  49™  38*. 52; 
III,    50m  35S.I2;   IV,    5im   3is.70;  V,    52m  28S.79;  VI,    53™ 
26s. 40;  VII,  54m  22s. 5 8.      Derive  the  equatorial  intervals  of 
the  various  lines  from  the  mean  line. 

Ans.    I,    —40s. 93;    II,      -27s. 17;    III,      -  I3S.62;    IV, 
-os.o8;  V,  +  I3S. 59;  VI,  +27S.38;  VII,  +  4OS.83. 

3.  While  observing  a  transit  of  the  star  24  Comae  (6  =  18° 
57')  clouds  interfered  so  that  observations  upon  the  first  and 
second  lines  of  the  reticle  were  missed.     The  observed  times 
of  transit  across  the  remaining    lines  were  as  follows:    III, 
8h  I3m20s.4o;  IV,  I3m348.93;  V,  i3m49s.2S;  VI,  I4mo38.88; 
VII,  I4m  i8s.io.      The  known  equatorial  interval  of  the  first 
line  is  —  40*. 86,  and  of  the  second  —  27s.3i.     Deduce  the 
time  of  transit  across  the  mean  of  the  seven  lines. 

Ans.   8h  I3m  34S.90. 

4.  Draw  two  diagrams  illustrating  the  geometric  relations 
from  which  formulae  (34)  and  (35)  of  §  96  are  derived. 

5.  The  following  ten  stars  were  observed  for  time  with  a 
Troughton  and  Simms  transit  at  Cornell  University  on  May 
23,   1896.      Given  the  partially  reduced  results  as  indicated 
below,  compute  the  correction  to  the  Howard  clock  (keeping 
mean  time)  with  which  the  observations  were  made. 


GEODETIC  ASTRONOMY.  §  133. 

Position  Corrected  Transit  Right  Ascension 

Star.  o.  of  Across  Reduced  to 

Lamp.  Mean  Line.*  Mean  Time. 

e  Virginis 11°  31'  W  8h   $6m  i8'.84  8h  48ra  22'.66 

43  Comae. . . . .......  28     24  W  g     06     18.93  8     58     22.48 

20  Can,  Ven 41     07  W  g     12     09.90  9     04     13.35 

£  Ursae  Maj.. . . 55     28  W  g     19    00.84  9     n     04.30 

Gr.  2001.  o 72     56  IV  9     22     47.60  9     14     50.19 

17  H.  Can.  Ven 37     43  E  g     29     23.87  9     21     26.84 

7?  Ursae  Maj 49     50  E  g     42     39.58  9     34    42.52 

V  Bootis 18     55  E  g     48     54.94  9     40     58.23 

ii  Bootis 27     53  E  g     55     37.63  9     47     40,83 

a  Drac 64     52  E  10    oo    45.68  9     52     48.27 

Ans.  By  the  method  of  §  104,  ATC  —  —  ;m  56S.73,  c  = 
+  O8.i7,  aw=  +  os.73,  and  aE  =  +os.3O. 

By  the  method  of  §  107,  ATC=  —  jm  $6*. 74,  ±0.02, 
aw  =  +  o8. 69,  aE  =  +  o8. 34,  c  —  +  os.  17. 

6.  The  following  ten  stars  were  observed  for  time  at 
Washington,  D.  C.  (0  —  38°  54'),  on  June  22,  1896,  with  a 
sidereal  chronometer.  Given  the  following  data,  compute 
the  chronometer  correction  on  local  sidereal  time: 

Position        Corrected  Transit  Right  Ascension 

Star.  5.  of  Across  Reduced  to 

Lamp.  Mean  Line.*  Mean  Time. 

3  Serpentis 5°  19'  E  i5h  iorn  03'. 96  i5h  iom  O4«.i4 

i  H.  Urs.  Min 67  46  E  15  13  32.01  15  13  30.38 

/w  Bootis. 37  44  E  15  20  37.07  15  20  36.65 

*  Draconis 59  20  E  15  22  41.21  15  22  40^29 

^'Bootis 41  ii  E  15  27  15.11  15  27  14.70 

C  Cor.  Bor.  seq....  36  58  W  15  35  30.18  15  35  30.68 

K  Serpentis 18  28  W  15  44  05  .86  15  44  06.54 

C  Urs.  Min 78  07  W  15  47  52-19  15  47  51.32 

e  Cor.  Bor 27  ii  W  15  53  19.42  15  53  19.92 

6  Draconis 58  51  W  15  59  59-56  15  59  59-92 

Ans.  By  the  method  of  §  104,  ATC=  +  os.o8,  c~  + 
O8.32,  as  =  +  os.6g,  and  aw  =•  +  os.82. 

By  the  method  of  §  107,  ATC  =  +  os.o8  ±  os.O3,  c  =  + 
O8.32,  aE  =  +  os.66,  and  aw  —  +  os.8o. 

*  Transit   corrected    for    diurnal    aberration,    pivot    inequality,    and 
inclination. 


§133-  QUESTIONS  AND   EXAMPLES.  149 

7.  Suppose  that  a  striding  level  carries  a  continuous 
graduation  of  one  hundred  divisions  each  one-twentieth  of  an 
inch  long,  and  that  each  division  represents  one  second  of  arc. 
By  about  how  much  does  the  arc  which  is  the  longitudinal 
section  of  the  upper  inner  surface  of  the  level  tube  depart 
from  the  chord  of  that  arc  joining  the  end  graduations  ? 

Ans    0.00030  inch. 


ISO  GEODETIC  ASTRONOMY.  §  134. 


CHAPTER   V. 

THE   ZENITH   TELESCOPE   AND  THE  DETERMINATION  OF 

LATITUDE. 

The  Principle  of  the  Zenith  Telescope. 
134.  The  zenith  distance  pf  a  star  when  on  the  meridian 
is  the  difference  between  the  latitude  of  the  station  of  obser- 
vation and  the  declination  of  the  star.  Hence  a  measurement 
of  the  meridional  zenith  distance  of  a  known  star  furnishes  a 
determination  of  the  latitude.  In  the  zenith  telescope,  or 
Horrebow-Talcott,  method  of  determining  the  latitude  there 
is  substituted  for  this  measurement  of  the  absolute  zenith 
distance  of  a  star  .the  measurement  of  the  small  difference  of 
zenith  distances  of  two  stars  culminating  *  at  about  the  same 
time  on  opposite  sides  of  the  zenith.  The  effect  of  this  sub- 
stitution is  the  attainment  of  a  much  higher  degree  of  pre- 
cision, arising  from  the  increased  accuracy  of  a  differential 
measurement,  in  general,  over  the  corresponding  absolute 
measurement;  from  the  elimination  of  the  use  of  a  graduated 
circlef  in  the  measurement;  and  from  the  fact  that  the  com- 
puted result  is  affected,  not  by  the  error  in  estimating  the 
absolute  value  of  the  astronomical  refraction,  but  simply  by 
the  error  in  estimating  the  very  small  difference  of  refraction 
of  two  stars  at  nearly  the  same  altitude. 

*  A  star  is  said  to  culminate  at  the  instant  when  it  crosses  the  meridian. 

\  The  zenith  telescope  carries  a  graduated  circle,  but  it  is  used  simply 
as  a  finder  or  setting  circle,  and  its  readings  do  not  enter  the  computed 
result. 


§  135-  THE   ZENITH   TELESCOPE.  I$I 

One  may  form  a  concrete  conception  of  the  relation 
between  the  latitude  and  the  measured  difference  of  zenith 
distance  as  follows:  Suppose  an  observer,  A,  measures  the 
difference  of  the  meridional  zenith  distances  of  two  stars  and 
finds  one  to  be  i°  farther  south  of  his  zenith  than  the  other 
is  north  of  it.  Suppose  that  another  observer,  By  is  stationed 
just  i'  due  north  of  A,  and  measures  the  difference  of  zenith 
distances  of  those  same  stars  at  the  same  times.  For  B  the 
southern  star  will  evidently  be  \'  farther  from  the  zenith  than 
for  A,  and  the  northern  star  i'  nearer  the  zenith.  Hence  B 
will  find  the  difference  of  the  zenith  distances  to  be  i°  02' '. 
Or,  a  given  change  in  the  position  of  the  observer,  along  a 
meridian,  produces  double  that  change  in  the  difference  of 
^zenith  distances  of  two  stars  which  culminate  on  opposite 
sides  of  the  zenith.  (Let  the  student  draw  a  figure,  in  the 
plane  of  the  meridian,  to  illustrate  this  paragraph.) 

Description  of  the  Zenith  Telescope. 

135.  Fig.  20  shows  a  zenith  telescope  which  is  the  prop- 
erty of  the  Coast  and  Geodetic  Survey. 

The  arm  A  turns  with  the  superstructure  of  the  instru- 
ment and  may  be  clamped  to  the  horizontal  circle,  which  is 
fixed  to  the  base.  At  B  and  B  are  two  stops  which  may  be 
clamped  to  the  circle  in  such  positions  that  the  telescope  will 
be  in  the  meridian  when  the  arm  A  is  in  contact  with  either 
of  them.  One  end  of  the  horizontal  axis  is  shown  at  C.  The 
striding  level  is  shown  at  D.  It  is  counterweighted  so  as  to 
make  it  balance  on  the  horizontal  axis.  By  means  of  the 
vernier  and  tangent  screw  at  E,  the  levels  FF  (called  latitude 
levels)  can  be  set  at  any  required  angle  with  the  telescope. 
These  levels  each  carry  a  2-mm.  graduation  of  50  divisions, 
numbered  continuously  from  one  end.  The  value  of  one 
division  is  about  1.5  seconds.  One  level  would  serve  the 


152  GEODETIC  ASTRONOMY.  §  135. 

purpose,  but  two  were  placed  upon  this  instrument  so  that 
increased  accuracy  might  be  secured  by  reading  both.  By 
means  of  the  clamp  at  G  and  the  tangent  screw  at  H,  operat- 
ing upon  the  sector  /,  the  telescope  may  be  brought  to  any 
desired  inclination. 

The  object-glass  has  a  clear  aperture  7.6  cm.  (=  3.0  in.) 
in  diameter,  and  its  focal  length  is  116.6  cm.  (=  45.9  in.). 
The  eyepiece  has  a  magnifying  power  of  100  diameters.  The 
focal  plane  of  the  object-glass  lies  in  the  rectangular  brass  box 
shown  aty.  The  micrometer  screw,  of  which  the  graduated 
head  is  shown  at  K,  controls  a  rectangular  brass  frame  sliding 
in  parallel  guides  within  this  box.  The  movable  line  with 
which  the  star  bisections  are  made  is  stretched  across  the 
sliding  frame.  While  in  use  the  object-glass  is  so  focused  as 
to  make  the  focal  plane  coincide  with  the  plane  in  which  this 
line  moves. 

To  facilitate  counting  the  whole  turns  of  the  micrometer 
screw  a  small  brass  strip  is  placed  in  one  side  of  the  field  of 
view  of  the  eyepiece  nearly  in  the  plane  of  the  micrometer 
line.  The  edge  of  the  strip  is  filed  into  notches  o.oi  in. 
apart.  The  pitch  of  the  screw  being  o.oi  in.,  the  micrometer 
line  appears  to  move  one  notch  along  this  comb  for  each  com- 
plete turn  of  the  screw.  The  whole  turns  are  thus  read  from 
the  comb,  and  the  fractions  are  read  from  the  head  of  the 
screw,  which  is  graduated  into  one  hundred  equal  divisions. 

In  Fig.  21,  drawn  in  a  vertical  plane  through  the  center 
of  the  telescope,  let  O  be  the  optical  center  *  of  the  object- 
glass.  Let  5  be  the  position  of  a  star.  The  star  image  is 
formed  at  the  focus  Ty  which  is  necessarily  in  the  line  SO 
produced.  If  the  star  is  to  appear  bisected,  the  micrometer 
line  must  be  placed  at  T.  If  another  star  later  occupies  the 

*  The  optical  center  of  a  lense  is  that  point  through  which  all  incident 
rays  pass  without  permanent  change  of  direction. 


§  136.  ADJUSTMENTS.  1 53 

position  S',  its  image  will  be  formed  at  T' ,  in  S'O  produced, 
and  to  make  a  bisection  the  micrometer  screw  must  be  turned 
until  the  micrometer  line  is  at  T' .  The  recorded  number  of 
turns  of  the  micrometer  screw  required  to  move  the  line  from 
T  to  T'  gives  a  measurement  of  the  linear  distance  TT' . 
For  the  small  angles  concerned  this  linear  distance  is  propor- 
tional to  the  angle  TOT' ,  the  equal  of  SOS' .  Hence  the 
observed  movement  of  the  micrometer  screw  gives  a  measure- 
ment of  the  difference  of  zenith  distances,  SOS' ,  of  the  two 
stars.  In  this  particular  instrument,  the  pitch  of  the  screw 
being  about  o.oi  in.  and  the  focal  length  OT  about  45.9  in., 

o.oi 
one  turn  of  the  screw  measures  an  angle  *  of  about  sin"1  , 

45-9 
or  about  40". 

The  more  common  form  of  zenith  telescope  differs  from 
the  one  here  shown  in  having  the  telescope  mounted  eccen- 
trically on  one  side  of  the  vertical  axis  instead  of  in  front  of 
it,  as  in  this  case;  in  having  a  clamp  which  acts  directly  upon 
the  horizontal  axis  in  the  place  of  the  clamp  at  G  acting  on 
the  sector  /;  and  in  having  only  one  latitude  level  instead  of 
two. 

Adjustments. 

136.  The  vertical  axis  must  be  made  truly  vertical.  In 
adjusting  and  using  the  instrument  it  will  be  found  conven- 
ient to  have  two  of  the  three  foot-screws  in  an  east  and  west 
direction.  The  vertical  axis  may  be  made  approximately 
vertical  by  use  of  the  plate  level,  if  there  is  one  on  the  instru- 
ment, and  the  final  adjustment  made  by  using  the  latitude 

*  The  value  of  one  turn  cannot  be  determined  with  sufficient  accuracy 
by  such  linear  measurements.  They  are  given  here  merely  to  illustrate 
the  principle  involved.  The  indirect  process  by  which  the  value  is  ordi- 
narily determined  will  be  found  described  in  §§  158-164. 


154  GEODETIC  ASTRONOMY.  §  138. 

level.  The  process  in  each  case  is  precisely  the  same  as  that 
of  using  the  unadjusted  plate  levels  of  an  engineer's  transit 
to  adjust  its  vertical  axis. 

The  horizontal  axis  must  be  perpendicular  to  the  vertical 
axis.  This  may  be  tested,  after  the  vertical  axis  has  been 
adjusted,  by  reading  the  striding  level  in  both  of  its  positions. 
If  the  horizontal  axis  is  inclined,  it  must  be  made  horizontal 
by  using  the  screws  which  change  the  angle  between  the  hori- 
zontal and  vertical  axes. 

137.  The  line  of  collimation  must  be  perpendicular  to  the 
horizontal  axis.      If  the  instrument  is  of  the  form  shown  in 
Fig.  20,  this  adjustment  may  be  made  as  for  an  astronomical 
transit  (§  86)  by  reversing  the  horizontal  axis  in  the  Ys.      If 
the  instrument  is  of  the  form  in  which  the  telescope  is  eccen- 
tric with  respect  to  the  vertical  axis,  the  method  of  making 
the  test  must  be  modified  accordingly.      It  may  be  made  as 
for  an  engineer's  transit,  but  using  two  fore  and  two  back 
points,  the  distance  apart  of  each  pair  of  points  being  made 
double  the  distance  from  the  vertical  axis  to  the  axis  of  the 
telescope.      Or,  a  single  pair  of  points  at  that  distance  apart 
may  be  used  and  the  horizontal  circle  trusted  to  determine 
when  the  instrument  has  been  turned   180°  in  azimuth.      If 
one  considers  the  allowable  limit  of  error  in  this  adjustment 
(see  §  167),  it  becomes  evident  that  a  telegraph  pole  or  small 
tree,    if   sufficiently   distant    from    the   instrument,    may    be 
assumed  to  be  of  a  diameter  equal  to  the  required  distance 
between  the  two  points.     Or,  a  single  point  at  a  known  dis- 
tance may  be  used  and  a  computed  allowance  made  on  the 
horizontal  circle  for  the  parallax  of  the  point  when  the  tele- 
scope is  changed  from  one  of  its  positions  to  the  other. 

138.  During  daylight  the  object-glass  should  be  carefully 
focused  on  the  most  distant  well-defined  object  available,  to 
insure  that  stars  may  be  seen  at  night.     A  neglect  to  do  this 


§  139-  ADJUSTMENTS.  155 

may  cause  the  observer,  especially  if  inexperienced,  much 
annoyance  while  he  is  trying  to  find  out  why  stars  for  which 
the  settings  are  properly  made  do  not  appear  in  the  telescope. 
At  the  first  opportunity  the  focus  should  be  tested  upon  a 
star.  When  once  the  focus  has  been  satisfactorily  adjusted 
at  a  station,  so  that  there  is  no  parallax,  it  should  not  again 
be  changed  at  that  station.  For  any  change  in  the  object- 
glass  focus  changes  the  angular  value  of  a  division  of  the 
micrometer.  It  is  well  to  clamp  the  slide  so  as  to  make  an 
accidental  change  of  focus  impossible. 

The  stops  on  the  horizontal  circle  must  be  set  so  that  when 
the  abutting  piece  is  in  contact  with  either  of  them  the  line 
of  collimation  is  in  the  meridian.  For  this  purpose,  and 
throughout  the  observations,  the  chronometer  correction  must 
be  known  roughly,  within  one  second,  say.  Set  the  telescope 
for  an  Ephemeris  star  which  culminates  well  to  the  northward 
of  the  zenith,  and  look  up  the  apparent  right  ascension  for  the 
date.  Follow  the  star  with  the  middle  vertical  line  of  the 
reticle,  at  first  with  the  horizontal  motion  free,  and  afterward 
using  the  tangent  screw  on  the  horizontal  circle,  until  the 
chronometer,  corrected  for  its  error,  indicates  that  the  star  is 
on  the  meridian.  Then  clamp  a  stop  in  place  against  the 
abutting  piece.  Repeat  for  the  other  stop,  using  a  star  which 
culminates  far  to  the  southward  of  the  zenith.  It  is  well  to 
test  the  setting  of  each  stop  again  by  an  observation  of 
another  star  before  commencing  latitude  observations. 

139.  The  movable  line,  attached  to  the  micrometer,  with 
which  pointings  are  to  be  made  must  be  truly  horizontal. 
This  adjustment  may  be  made,  at  least  approximately,  in 
daylight  after  the  other  adjustments.  Point,  with  the  mov- 
able line,  upon  a  distant  well-defined  object,  with  the  image 
of  that  object  near  the  apparent  right-hand  side  of  the  field  of 
the  eyepiece.  Shift  the  image  to  the  apparent  left-hand  side 


156  GEODETIC  ASTRONOMY.  §  140. 

of  the  field  by  turning  the  instrument  about  its  vertical  axis. 
If  the  bisection  is  not  still  perfect,  half  the  correction  should 
be  made  with  the  micrometer  and  half  with  the  slow-motion 
screws  which  rotate  the  whole  eyepiece  and  reticle  about  the 
axis  of  figure  of  the  telescope.  The  adjustment  should  be 
carefully  tested  at  night  after  setting  the  stops,  by  taking  a 
series  of  pointings  upon  a  slow-moving  star  as  it  crosses  the 
field  with  the  telescope  in  the  meridian.  If  the  adjustment 
is  perfect  the  mean  reading  of  the  micrometer  before  the  star 
reaches  the  middle  of  the  field  should  agree  with  its  mean 
reading  after  passing  the  middle,  except  for  the  accidental 
errors  of  pointing.  It  is  especially  important  to  make  this 
adjustment  carefully,  for  the  tendency  of  any  inclination  is  to 
introduce  a  constant  error  into  the  computed  values  of  the 
latitude. 

The  Observing  List. 

140.  Before  commencing  the  observations  at  a  station,  an 
observing  list  should  be  prepared,  showing,  for  each  star  to 
be  observed,  its  catalogue  number  or  its  name,  its  magnitude, 
mean*  right  ascension  and  declination  (at  the  beginning  of 
the  year),  zenith  distance,  whether  it  culminates  north  or 
south  of  the  zenith;  and  for  each  pair  the  setting  of  the  ver- 
tical circle  (the  mean  of  the  two  zenith  distances),  the  differ- 
ence of  the  zenith  distances  with  its  algebraic  sign  as  given  by 
formula  (54);  and  finally  the  micrometer  comb  setting  for  each 
star.  For  the  purposes  of  the  observing  list  the  right  ascen- 
sions to  within  one  second  of  time,  and  the  declinations  and 
derived  quantities  within  one  minute  of  arc,  are  sufficiently 
accurate.  If  the  micrometer  comb  reading  is  one  minute  per 
notch,  and  the  middle  notch  is  called  20,  the  comb  setting  for 

*  For  the  definition  of  the  mean  place  of  a  star  see  §§  37,  39. 


§141.  OBSERVING   LIST.  1 57 

one  star  is  20  +  the  half-difference  of  zenith  distances  for  one 
star,  and  20  —  that  half-difference  for  the  other  star  of  a  pair. 

The  requisites  for  a  pair  of  stars  for  this  list  are  that  their 
right  ascensions  shall  not  differ  by  more  than  2Om,  to  avoid 
too  great  errors  from  instability  in  the  relative  positions  of 
different  parts  of  the  instrument;  nor  by  less  than  im,  that 
interval  being  required  to  take  the  readings  upon  the  first  star 
and  prepare  for  the  second  star  of  a  pair;  that  their  difference 
of  zenith  distances  shall  not  exceed  half  the  length  of  the 
micrometer  comb,  20'  for  the  usual  type  of  instrument;  that 
each  star  shall  be  bright  enough  to  be  seen  distinctly — not 
fainter  than  the  seventh  magnitude  for  the  instruments  here 
described;  and  that  no  zenith  distance  shall  exceed  35°,  to 
guard  against  too  great  an  uncertainty  in  the  refraction.  The 
selection  of  a  series  of  such  pairs  from  the  stars  of  a  catalogue 
requires  much  time  and  patience. 

The  total  range  of  the  list  in  right  ascension  is  governed 
by  the  hours  of  darkness  on  the  proposed  dates  of  observa- 
tion, and  by  the  convenience  of  the  observer.  The  third  of 
the  above  conditions  may  perhaps  be  used  more  conveniently 
in  this  form:  the  sum  of  the  two  declinations  must  not  differ 
from  twice  the  latitude  by  more  than  20'.  To  prepare  the 
list  the  latitude  of  the  station  should  be  known  within  a 
minute.  It  may  possibly  be  secured  from  a  map;  if  not,  then 
from  a  sextant  observation  of  the  Sun,  or  from  an  observation 
of  the  meridional  zenith  distance  of  a  star  with  the  finder 
circle  of  the  zenith  telescope. 

141.  The  stars  selected  should  be  such  that  their  com- 
puted mean  places  may  be  made  to  depend  in  each  individual 
case  upon  observations  at  several  different  observatories. 
The  declination  of  a  star  as  derived  from  observations  at  a 
single  observatory  will  not  in  general  be  sufficiently  accurate 
for  the  purpose  in  hand. 


1 58  GEODETIC  ASTRONOMY.  §  141. 

If  the  observer  knows  that  in  making  the  computation  an 
ample  collection  of  catalogues  *  of  original  observations  at 
each  of  various  observatories  will  be  available,  he  may  select 
all  the  suitable  pairs  he  can  find  in  any  extensive  list  of  stars, 
say  the  British  Association  Catalogue,  or  any  of  the  Green- 
wich Catalogues,  and  trust  to  finding  afterward  in  the  various 
catalogues  a  sufficient  number  of  observations  upon  each  star 
at  various  observatories  to  give  an  accurate  determination  of 
its  place.  This  is  the  usual  procedure  in  the  Coast  and 
Geodetic  Survey.  A  computer  in  the  office  at  Washington 
calculates  the  declinations  of  the  stars  which  have  been 
observed  for  latitude  by  bringing  together  in  a  least  square 
adjustment  all  the  observations  upon  each  star  that  he  finds 
in  his  large  collection  of  star  catalogues. 

If  the  necessary  collection  of  catalogues  of  original  obser- 
vations is  not  known  to  be  available,  the  observing  list  had 
best  be  made  up  from  star  lists  in  which  the  declinations  given 
are  the  result  of  the  compilation  and  computation  of  original 
observations  at  various  observatories  as  outlined  above. 
Among  such  available  lists  of  mean  places  are  those  in  the 
various  national  ephemerides;  Preston's  Sandwich  Island  List 
in  the  Coast  and  Geodetic  Survey  Report  for  1888,  Appendix 
No.  14,  pp.  511-523;  the  list  given  in  Appendix  No.  7,  pp. 
83-129,  C.  &  G.  S.  Report  for  1876;  and  Boss'  list  in  the  re- 
port of  the  Survey  of  the  Northern  Boundary  from  the  Lake 
of  the  Woods  to  the  Rocky  Mountains,  pp.  592-615.  These 
lists  are  given  in  about  the  order  of  the  accuracy  of  their  star 
places  when  reduced  to  the  present  time,  the  more  recently 
computed  places  being  more  accurate,  if  other  conditions  are 
about  the  same.  In  the  report  of  the  Mexican  Boundary 
Survey  of  1892-93,  which  is  about  to  be  published,  there  will 

*  An  indication  of  what  is  meant  by  "an  ample  collection  of  catalogues  " 
may  be  gained  by  reading  §  37. 


§  142  OBSERVING   LIST.  1 59 

be  given  an  unusually  accurate  list  of  declinations  prepared 
for  that  survey  by  Prof.  T.  H.  Safford. 

142.  If  an  observer  finds  difficulty  in  securing  a  sufficient 
number  of  pairs  from  the  available  lists  of  accurately  com- 
puted places,  he  may  extend  his  list  in  two  ways.  Firstly, 
and  preferably,  if  he  has  a  good  instrument,  by  extending  the 
limits  given  in  §  140.  The  limit  of  difference  of  right  ascen- 
sions may  not  safely  be  extended  much ;  the  limit  of  difference 
of  zenith  distances  may  be  extended  to  the  full  length  of  the 
micrometer  comb,  say  40" ;  and  zenith  distances  as  great  as 
45°  may  be  allowed.  Secondly,  he  may  have  recourse  in  his 
extremity  to  a  catalogue  of  original  observations  at  the 
Greenwich  observatory  for  enough  pairs  to  complete  his  list. 
The  number  of  pairs  required  may  be  estimated  by  the  con- 
siderations dealt  with  in  §  169. 

In  the  list  of  pairs  resulting  directly  from  the  search  in  the 
star  catalogues  there  will  be  many  pairs  which  overlap  in  time. 
A  feasible  observing  list  may  be  formed  by  omitting  such 
pairs  that  among  the  remainder  the  shortest  interval  between 
the  last  star  of  one  pair  and  the  first  star  of  the  next  shall  not 
be  less  than  2m.  In  that  interval  a  rapid  observer  can  finish 
the  readings  upon  one  pair,  set  and  be  ready  for  the  next, 
under  favorable  circumstances.  The  omitted  pairs  may  be 
included  in  a  list  prepared  for  the  second  or  third  night  of 
observation  if  one  uses  the  second  plan  outlined  in  §  169. 
Also,  it  will  frequently  be  found  that  the  same  star  occurs  in 
two  or  more  different  pairs.  Such  pairs  may  be  treated  like 
those  which  overlap  in  time,  or  the  three  or  more  stars  form- 
ing what  might  be  called  a  compound  pair  may  all  be  observed 
at  one  setting  of  the  telescope  and  then  treated  in  the  com- 
putation as  two  or  more  separate,  but  not  independent,  pairs. 

It  is  desirable  to  so  select  the  pairs  that  the  algebraic  sum 
of  all  the  differences  of  zenith  distances  for  a  station  shall  be 


l6b  GEODETIC  ASTRONOMY.  §  143. 

nearly  zero,  so  as  to  make  the  computed  latitude  for  the 
station  nearly  free  from  any  effect  of  error  in  the  mean  value 
of  the  micrometer  screw. 

Directions  for  Observing. 

143.  The  instrument  being  adjusted,  set  the  vertical  circle 
to  read  the  mean  zenith  distance,  or  "  circle  setting"  as 
marked  in  the  observing  list,  of  the  first  pair.  Direct  the 
telescope  to  that  side  of  the  zenith  on  which  the  first  star  of 
the  pair  will  culminate.  Put  the  bubble  of  the  latitude  level 
nearly  in  the  middle  of  the  tube  by  using  the  tangent  screw 
which  changes  the  inclination  of  the  telescope.  Place  the 
micrometer  thread  at  that  part  of  the  comb  at  which  the  star 
is  expected,  as  shown  by  the  observing  list.  Watch  the 
chronometer  to  keep  posted  as  to  when  the  star  should 
appear.  When  the  star  enters  the  field  place  the  micrometer 
thread  approximately  upon  it.  and  center  the  eyepiece  over 
the  thread.  As  soon  as  the  star  comes  within  the  limits  indi- 
cated by  the  vertical  lines  of  the  reticle  bisect  it  carefully. 
As  the  star  moves  along  watch  the  bisection  and  correct  it  if 
any  error  can  be  detected.  Because  of  momentary  changes 
in  the  refraction,  the  star  will  usually  be  seen  to  move  along 
the  thread  with  an  irregular  motion,  now  partly  above  it,  now 
partly  below.  The  mean  position  of  the  star  is  to  be  covered 
by  the  line.  An  attempt  is  being  made  to  secure  a  result 
which  is  to  be  in  error  by  much  less  than  the  apparent  width 
of  the  thread,  hence  too  much  care  cannot  be  bestowed  upon 
the  bisection.  It  is  possible,  but  not  advisable,  to  make 
several  bisections  of  the  star  while  it  is  passing  across  the 
field.  As  soon  as  the  star  reaches  the  middle  vertical  line  of 
the  reticle  read  off  promptly  from  the  comb  the  whole  turns 
of  the  micrometer,  read  the  level,  and  then  the  fraction  of  a 
micrometer  turn,  in  divisions,  from  the  micrometer  head.  Set 


§145'  DIRECTIONS   FOR    OBSERVING.  l6l 

promptly  for  the  next  star,  even  though  it  is  not  expected 
soon.  In  setting  for  the  second  star  of  a  pair  all  that  is 
necessary  is  to  reverse  the  instrument  in  azimuth  and  set  the 
micrometer  thread  to  a  new  position. 

144.  The  instrument  must  be  manipulated  as  carefully  as 
possible.      Especial  care  should   be   taken   in    handling   the 
micrometer  screw,  as  any  longitudinal  force  applied  to  it  pro- 
duces a  flexure  of  the   telescope  which  tends  to  enter  the 
result  directly  as  an  error.    The  last  motion  of  the  micrometer 
head  in  making  a  bisection  should  always  be  in   the  same 
direction  (preferably  that  in  which  the  screw  acts  positively 
against  its  opposing  spring),  to  insure  that  any  lost  motion  is 
always  taken  up  in  one  direction.     The  bubble  should  be  read 
promptly,  so  as  to  give  it  as  little  time  as  possible  to  change 
its  position  after  the  bisection.     The  desired  reading  is  that 
at  which  it  stood  at  the  instant  of  bisection.     Avoid  carefully 
any  heating  of  the  level  by  putting  the  reading  lamp,  warm 
breath,  or  face  any  nearer  to  it  than  necessary.     During  the 
observation  of  a  pair  the  tangent  screw  of  the  setting  circle 
must  not  be  touched,  for  the  angle  between  the  level  and 
telescope  must  be  kept  constant.     If  it  is  necessary  to  relevel, 
to  keep  the  bubble  within  reading  limits,    use  the    tangent 
screw  which  changes  the  inclination  of  the  telescope.      Even 
this  may  introduce  an  error,  due  to  a  change  in  the  flexure  of 
the  telescope,  and  should  be  avoided  if  possible. 

145.  For    first-class  observing   it  is  desirable  to  have    a 
recorder.      He    may    count    seconds    from    the    face    of    the 
chronometer  for  a  minute  before  culmination  in  such  a  way 
as  to  indicate  when  the  star  is  to  culminate  according  to  the 
right  ascension  given  on  the  observing  list,  taking  the  known 
chronometer  correction  into  account.     Such  counting  aloud 
serves  a  double  purpose.      It  is  a  warning  to  be  ready  and 
indicates  where  to  look  for  the  star  if  it  is  faint  and  difficult 


1 62 


GEODETIC  ASTRONOMY. 


§I46. 


to  find.  It  also  gives  for  each  star  a  rough  check  upon  the 
position  of  the  azimuth  stops  and  warns  the  observer  when 
they  need  readjustment.  It  is  only  a  rough  check,  because 
the  observing  list  gives  mean  right  ascensions  (for  the  begin- 
ning of  the  year)  instead  of  apparent  right  ascensions  for  the 
date.  But  in  view  of  §  167  it  is  sufficiently  accurate.  The 
observer  can  easily  make  allowance  for  the  fact  that  all  stars 
will  appear  to  be  fast  or  slow  according  to  the  observing  list 
by  about  the  same  interval,  o8  to  5"  (the  difference  between 
the  mean  and  the  apparent  place).  If  a  star  cannot  be 
observed  upon  the  middle  line,  on  account  of  temporary 
interference  by  clouds  or  tardiness  in  preparing  for  the  obser- 
vation, observe  it  anywhere  within  the  safe  limits  of  the  field 
as  indicated  by  the  vertical  lines  of  the  reticle  and  record  the 
chronometer  time  of  observation. 


EXAMPLE     OF     RECORD. 

146.   Station — No.  8,  near  San  Bernardino  Ranch,  Arizona. 
Instrument — Wurdemann  Zenith  Telescope  No.  20. 
Observer— J.  F.  H. 
Dale— August  9,  1892. 


Micrometer. 

Level. 

No.  of 

Star  No. 

N.  or 

Pair. 

B.  A.  C. 

S. 

Turns. 

Divisions. 

N. 

S. 

Remarks. 

OQ 

7528 

S. 

22 

82.0 

19.9 

54-9 

Sky  perfectly  clear. 

7544 

N. 

16 

98.9 

56.0 

20.9 

Chronometer  21*  fast. 

7566 

N. 

24 

71.2 

52.9 

17.9 

91 

7586 

S. 

13 

68.9 

17-9 

52.8 

7631 

N. 

30 

29.0 

52-5 

17.6 

93 

7662 

S. 

9 

13-0 

I6.4 

Sr-7 

§  148.  FORMULAE.  163 


Derivation  of  Formula. 

147.  Let  C  and  Cx  be  the  true  meridional  zenith  distance, 
and  d  and  6'  the  declination,  of  the  south  and  north  star  of  a 
pair,  respectively.     Then  the  latitude  of  the  station  is 

*=i(*+*')  +  KC-C') (54) 

Let  the  student  draw  the  figure  and  prove  this  formula. 

Let  (z  —  z')  be  the  observed  difference  of  zenith  distances 
of  the  two  stars,  the  primed  letter  referring  to  the  north  star, 
(z  —  ^)  is  in  terms  of  the  observed  micrometer  readings 
=  (M —  M')r,  in  which  M  and  Mr  are  the  micrometer  read- 
ing upon  the  south  and  north  star,  respectively,  expressed  in 
turns,  and  r  is  the  angular  value  of  one  turn.  Before  this 
observed  difference  of  zenith  distances  may  be  used  in  (54)  it 
must  be  corrected  for  the  inclination  of  the  vertical  axis  as 
given  by  the  level  readings,  for  refraction,  and  for  reduction 
to  the  meridian  if  either  star  is  observed  off  the  meridian. 

148.  Let  d  be  the  value  of  one  division  of  the  latitude 
level.      Let  n  and  s  be  the  north  and  south  reading,  respec- 
tively, of  the  level  for  the  south  star,  and  n'  and  s'  the  same 
for  the  north  star.     Then,  if  the  level  tube  carries  a  graduation 
of  which  the  numbering  increases  each  way  from  the  middle, 
the  inclination  of  the  vertical  axis,  considered  positive  if  the 
upper  end  is  too  far  south,  is 

jK«  +  *!)-(*+*')i (55) 

If  the  level  tube  carries  a  graduation  which  is  numbered 


UNIVERSITY 
sSLCAUFOaH^ 


I&J.  GEODETIC  ASTRONOMY.  §  149. 

continuously  from  one  end  to  the  other,  with  the  zero  nearest 
the  eyepiece,  the  inclination  of  the  vertical  axis  is 


'••'    ;;V  K«'  +  *')-(«  +  *)}  .....    (56) 

If  the  zero  is  nearest  the  object-glass  the  algebraic  sign  must 
be  changed  from  that  given  above. 

The  inclination  of  the  vertical  axis  makes  the  south  zenith 
distance  too  small  by  the  amount  indicated  by  (55)  or  (56) 
and  the  north  too  large  by  the  same  amount.  Hence  the 

d 

correction  to  \(z  —  z')  is  —  \(n  -f-  «')  —  (s  -f-  /)},  or  the  corre- 

4 

sponding  expression  (56). 

149.  The  refraction  makes  each  apparent  zenith  distance 
too  small.  If  R  and  R'  represent  the  refraction  for  the  south 
and  north  star,  respectively,  the  correction  to  (z  —  z')  is 
(R  -  R'\  and  to  J(*  -  /)  is  ±(R  -  R'). 

Let  m  be  the  correction  to  the  apparent  zenith  distance 
of  a  south  star  observed  slightly  off  the  meridian  to  reduce  it 
to  what  it  was  when  on  the  meridian,  and  m'  the  correspond- 
ing reduction  for  a  north  star  observed  off  the  meridian.  The 
correction  to  \(z  —  2')  will  then  be  J(/#  —  m').  m  and  m'  are 
of  course  zero  in  the  normal  case,  when  the  observation  is 
made  in  the  meridian. 

Formula  (54)  may  now  be  written,  for  an  instrument  with 
the  level  graduated  both  ways  from  the  middle, 


0  = 

*0  +  ..     (57) 


§!$!.  FORMULAE.  165 

This  is  the  working  formula  for  the  computation,  but  the 
values  of  the  last  two  terms  may  be  conveniently  tabulated. 

150.  The  difference  R  —  R  being  very  small,  the  variation 
of  the  state  of  the  atmosphere  at  the  time  of  observation  from 
its  mean   state   (see   refraction   tables,  §§  294-297)   may  be 
neglected,  except  for  stations  at  high  altitudes.     It  has  been 
shown,  by  the  investigations  of  the  laws  of  refraction  which 
have  been  referred  to  in  §§  67-69,  that  this  differential  refrac- 
tion, for  the  mean  state  of  the  atmosphere,  is,  with  sufficient 
accuracy  for  the  present  purpose, 

R-R'=  $7". 7  sin  (*  —  *>)  sec*  *.       .     .     (58) 

By  computation  from  this  formula  the  value  of  the  term 
%(R  —  Rf)  of  formula  (57)  has  been  tabulated,  in  §  304,  for 
the  arguments  %(z  —  z'}  as  directly  observed  With  the  microm- 
eter, and  the  zenith  distance. 

If  the  station  is  so  far  above  sea-level  that  the  mean 
barometric  pressure  is  less  than  y9^,  say,  of  760  mm.  (§  294), 
the  mean  pressure  at  sea-level,  it  is  necessary  to  take  this  fact 
into  account  by  diminishing  the  values  of  the  differential 
refraction  given  in  §  304  in  the  ratio  of  the  mean  pressures. 
That  is,  if  the  ,mean  pressure  is  10$  less  than  at  sea-level 
diminish  the  values  of  §  304  by  io#;  if  20$  less  subtract  20$, 
and  so  on.  Inspection  of  the  table  shows  that  this  allowance 
need  only  be  made  roughly,  since  the  tabular  values  are  small. 

151.  The  value  of  — ,  and  of  its  equal  — ,  is  tabulated  in 

§  305.  The  table  gives  directly  the  correction  to  the  latitude 
for  any  case  of  a  star  observed  off  the  meridian,  but  within 
one  minute  of  it.  If  both  stars  of  a  pair  are  observed  off  the 
meridian  two  such  corrections  must  be  applied,  one  for  each 
star.  For  the  difficult  derivation  of  the  formula  from  which 


i66 


GEODETIC  ASTRONOMY. 


§   152- 


this  table  is  computed  see  Chauvenet's  Astronomy,  vol.  II. 
pp.  346,  347;  or  Doolittle's  Practical  Astronomy,  pp.  505, 
$06. 


EXAMPLE   OF   COMPUTATION. 

152.    Station — No.  8,  near  San  Bernardino  Ranch,  Arizona. 
Instrument — Wurdemann  Zenith  Telescope  No.  20. 
Observer— J.  F.  H. 
Date — August  9,  1892. 

[Left-hand page  of  Computation.'} 


o 

1. 

Micrometer, 
i  turn  =  ioo  div. 

Level, 
i  div.  =  i".28. 

i 

<u    . 

3    . 

C/5 

.S  rt 

Declination. 

n'T 

>5**J 

VH 

Level 

JO  to 

c5ed 

Reading. 

Diff.  Z.  D. 

N. 

S. 

Corr.  in 

«Q 

* 

in 

fc 

Div. 

» 

t.        d. 

t.        d. 

7528 

S. 

22       82.0 

19.9 

54-9 

19°  46'  48".  62 

7544 

N. 

16    98.9 

+  5    83.1 

56.0 

20.9 

+  0.52 



42     47     05    .83 

7S66 

N. 

24    71.2 

52.9 

17.9 

37     47     26    .64 

91 

7586 

S. 

13    68.9 

—  ii    02.3 

17.9 

52.8 

4~  0.03 



25    °3     55    -38 

93 

7631 
7662 

N. 
S. 

30    29.0 
9     13-0 

52.5 
16.4 

17.6 

4-0.50 

55     17     23    -93 
7     44     26    .71 

[Right-hand page  of  Computation.] 


Sum  and  Mean 
of  Declinations. 

Corrections. 

Latitude. 

Remarks. 

Micrometer. 

Level. 

Refraction. 

Meridian. 

62°  33'  54"-45 
31     16     57   .22 

4-  3'  oi".o5 

+  o".67 

4-  o".o4 



31°  19'  58".98 

62       51       22     .02 

31     25     41    .01 

-   5    42  .26 

-f  o   .04 

—  o   .08 



58   .71 

63    oi     50  .64 
3i     3°     55   -3» 

—  10    57   .01 

+  o   .64 

-  o   .18 



58   .77 

§  153-  EXPLANATION  OF  COMPUTATION. 


Explanation  of  Computation. 

153.  The  first  seven  columns  of  the  computation  need 
no  explanation.  The  eighth  column  gives  the  values  of 
i{(»'  +  -0  —  (»  +  s)\,  §  148,  the  level  tube  of  this  instru- 
ment being  graduated  continuously  from  end  to  end  with  the 
zero  nearest  the  eyepiece.  The  ninth  column  gives  the 
meridian  distance  of  such  stars  as  were  not  observed  upon  the 
meridian.  It  is  the  hour-angle  of  the  star  expressed  in 
seconds  of  time.  To  obtain  the  apparent  right  ascension 
within  one  second,  which  is  sufficiently  accurate  for  the 
determination  of  the  required  hour-angle  for  the  present  pur- 
pose, proceed  as  follows:  Select  a  star  from  the  mean  place 
list  of  the  Ephemeris  which  has  nearly  the  same  right  ascen- 
sion and  declination  as  the  star  in  hand.  Compare  its  mean 
right  ascension  with  its  apparent  right  ascension  for  the  date 
and  assume  that  the  corresponding  change  for  the  star  in  hand 
is  the  same. 

The  declinations  given  in  column  ten  are  those  resulting 
from  the  apparent  place  computation  made  as  indicated  in 
§§  46-49,  or  from  the  Ephemeris  by  interpolation  in  case  the 
star  is  one  of  which  the  apparent  place  is  there  given. 

The  computation  of  the  values  in  the  second  column  of 
the  right-hand  page  of  the  computation  is  facilitated,  if  there 
are  many  observations,  by  first  constructing  a  table  giving  10, 
20,  30,  etc.,  turns  of  the  micrometer  reduced  to  arc  by  multi- 
plying by  — :  then  of  I,  2,  3,  4,  5,  6,  7,  8,  9  turns  reduced  to 

arc:  of  10,  20,  30,  40,  50,  60,  70,  80,  90  divisions  thus 
reduced  to  arc:  of  I,  2,  3,  ...  etc.:  and  of  o.  I,  0.2,  0.3, 
.  .  .  etc.  Such  a  table  reduces  the  multiplication  process 
otherwise  required  to  a  process  of  adding  five  tabular  quan- 
tities. 


l68  GEODETIC  ASTRONOMY.  §  154. 

The  corrections  for  refraction  were  obtained  by  subtract- 
ing 20$  from  the  values  of  §  304,  the  barometric  pressure 
being  only  about  four-fifths  as  great  at  San  Bernardino  as  at 
sea-level. 

The  latitude  is  obtained  from  each  pair  by  adding  the 
various  corrections  algebraically  to  the  mean  declination,  as 
indicated  in  formula  (57). 

With  sufficient  accuracy  for  some  purposes  the  indiscrimi- 
nate mean  of  all  the  individual  values  may  be  taken  as  the 
final  value  of  the  latitude.  If  the  best,  or  most  probable, 
value  is  desired  the  procedure  outlined  below  must  be  fol- 
lowed. 

Combination  of  Individual  Results  by  Least  Squares. 

154.  Let  us  first  deal  with  the  simplest  case.  Namely, 
let  it  be  supposed  that  /  separate  pairs  have  been  observed 
on  each  of  ri  nights  at  a  station,  each  pair  being  observed  on 
every  night.  For  this  case  it  will  be  found  that  the  indis- 
criminate mean  is,  after  all,  the  most  probable  value  of  the 
latitude,  but  the  principles  developed  will  be  found  useful  in 
dealing  with  other  more  difficult  cases  in  which  this  is  not  true. 

The  differences  A  obtained  by  subtracting  the  mean  result 
for  any  one  pair  from  the  result  on  each  separate  night  for 
that  pair  are  evidently  independent  of  errors  of  declination. 
We  may  compute  from  these  differences,  or  residuals,  the 
probable  error  of  a  single  observation  e.  This  error  of 
observation  includes  the  observer  s  errors,  instrumental  errors, 
and  all  external  errors  except  the  errors  of  the  assumed 
declinations.  Then  by  least  squares 


e  = 


*This  square  bracket  [  j  is  here  used  to  indicate  summation,  as  it  fre- 
quently is  in  text-books  on  least  squares. 


§  154-  COMBINATION  OF  RESULTS.  l6g 

in  which  \_A A\  stands  for  the  sum  of  the  squares  of  all  the 
residuals  A  obtained  from  all  the  pairs. 

The  probable  error  ep  of  the  mean  result  from  any  one 
pair  may  also  be  computed  from  the  observations  by  the 
formula 


(6o) 


in  which  v  is  .the  residual  obtained  by  subtracting  the  mean 
result  for  the  station  from  the  mean  result  for  each  pair. 
There  are  /  such  residuals.  \yv\  stands  for  the  sum  of  the 
squares  of  these  residuals. 

Let  eB  be  the  probable  error  of  the  mean  of  the  two 
declinations.  ep  evidently  includes  the  declination  errors  of 
the  two  stars  of  a  pair.  From  the  ordinary  law  of  transmis- 
sion of  accidental  errors 


Whence 


e$  may  thus  be  obtained,  from  the  observations  for  lati- 
tude, by  substituting  the  values  e  and  ep  computed  by  (59) 
and  (60)  in  (62). 

nf  being  the  same  for  all  pairs,  it  is  evident  from  (61)  that 
the  means  from  the  various  pairs  have  equal  weight.  The 
most  probable  value  for  the  latitude  is  then  the  indiscriminate 
mean  of  the  results  from  the  separate  pairs,  or,  what  is 
numerically  the  same  in  this  case,  the  indiscriminate  mean  of 
all  the  individual  results  for  latitude. 


I?O  GEODETIC  ASTRONOMY.  §  I55» 

The  probable  error  of  the  final  result  for  latitude  is 


(63) 


155.  The  simple  case  just  treated  seldom  occurs  in  prac- 
tice. Observations  upon  certain  pairs  are  missed  on  some  of 
the  nights  by  accident  or  by  cloud  interference;  work  may 
be  entirely  stopped  by  clouds  after  half  the  observations  of  an 
evening  have  been  made;  or  on  the  later  evenings  of  a  series 
the  observer  may  purposely,  with  a  view  to  more  effectual 
elimination  of  declination  errors,  include  in  his  observing  list 
certain  pairs  which  have  not  before  been  observed  at  that 
station,  in  the  place  of  pairs  which  have  already  been  observed 
once  or  more. 

In  the  usual  case,  then,  a  total  of  p  pairs  are  observed, 
pair  No.  I  being  observed  nl  times  (i.e.,  on  nl  nights),  pair 
No.  2  7za  times,  .  .  .  ,  and  the  total  number  of  observations 
is  n  =  nl  -{-  n^  -\-  nz  .  .  . 

By  the  same  reasoning  as  before  we  have,  by  the  ordinary 
least  square  formula,  the  probable  error  of  a  single  observa- 
tion 


=      Ao.455) 

"  V    »- 


P 


To  obtain  the  probable  error  ep  of  the  mean  result  from 
any  one  pair  with  rigid  exactness  it  is  necessary  to  take  into 
account  the  fact  that  different  pairs  must  now  be  given 
different  weights,  since  some  are  observed  more  times  than 
others.  To  do  this  would  make  the  computation  consider- 
ably longer  than  is  otherwise  necessary.  Fortunately,  inves- 
tigation of  the  numerical  values  concerned  shows  that  the 
results  are  abundantly  accurate  if  formula  (60)  is  here  used 


§  1 56.  COMBINATION  OF  RESULTS.  I? I 

and  the  fact  that  the  pairs  are  of  unequal  weight  neglected  in 
deriving  ep.     With  sufficient  accuracy,  then, 


(65) 


According  to  the  laws  of  accidental  errors  the  probable 
errors  of  the  mean  results  from  the  separate  pairs  are 


The  values  ep.lt  ep.^  .  .  .  differ  from  each  other  because  of 
the  various  values  of  «,,  n^  naJ  .  .  . 

For  use  in  deriving  e&  an  average  value  of  the  second  term 
under  the  radical  must  be  obtained.  Again  neglecting  the 
unequal  weights  of  the  pairs,  this  average  value,  which  will  be 
called  e%  is 


+«  '  +  »    '  *  V 


Corresponding  to  (62),  there  is  now  obtained 

(68) 


from  which  es  may  be  computed  from  the  latitude  observa- 
tions. 

156.  The  proper  weights  for  the  mean  results  from  the 
separate  pairs  are  inversely  proportional  to  the  squares  of 
their  probable  errors.  Hence  these  weights  wlt  w^  wt,  .  .  .  , 
are  proportional  to 

•  •  (69) 


1/2  GEODETIC  ASTRONOMY.  §  156. 

and  may  now  be    computed  from  the    known    values  of  e& 
and  e. 

The  most  probable  value  00  for  the  latitude  of  the  station 
is  the  weighted  mean  of  the  mean  results  from  the  various 
pairs,  or 

.    .  _   |>0] 

'        ' 


in  which  0,  is  the  mean  result  from  the  first  pair,  0a  from  the 
second  pair,  and  so  on. 

Also,  the  probable  error  of  this  result  is 


/^455)^3 

'  V  (/  -  ow  ' 


in  which  [wz/  a]  stands  for  the  sum  of  the  products  of  the 
weight  for  each  pair  into  the  square  of  the  residual  obtained 
by  subtracting  00  from  the  mean  result  for  that  pair,  and  [w] 
is  the  sum  of  the  weights. 

In  case  two  north  stars  are  observed  in  connection  with 
the  same  south  star,  or  vice  versa,  and  the  computation  is 
made  as  if  two  independent  pairs  had  been  observed,  the 
weight  of  each  of  these  pairs  as  given  by  (69)  should  be  mul- 
tiplied by  \  to  take  account  of  the  fact  that  they  are  but 
partially  independent.  Similarly  if  three  north  stars  have 
been  observed  in  connection  with  the  same  south  star  the 
weights  from  (69)  for  each  of  the  three  resulting  pairs  should 
be  multiplied  by  J.* 

If,  however,  a  given  north  star  is  observed  in  connection 
with  a  certain  south  star  on  a  certain  night  or  nights,  and 

*  Coast  and  Geodetic  Survey  Report,  1880,  p.  255  ;  or  Professional 
Papers  of  the  Corps  of  Engineers,  No.  24  (Lake  Survey  Triangulation),  p. 
625. 


§157-  COMBINATION  OF  RESULTS.  173 

on  a  certain  other  night  or  nights  is  observed  in  connection 
with  some  other  south  star,  the  case  is  different,  and  the  com- 
putation is  sufficiently  accurate,  though  not  exact,  if  each  of 
such  pairs  is  given  the  full  weight  resulting  from  (69). 

If  very  few  pairs  are  observed  more  than  once  at  a  sta- 
tion the  determination  of  es  from  the  latitude  observations 
obviously  fails,  and  it  must  be  estimated  in  some  other  way  — 
from  the  star  catalogues,  for  example. 

157.  As  an  example  of  the  application  of  formulae  (64)  to 
(71),  the  process  of  combining  the  various  values  for  the  lati- 
tude of  station  No.  8  on  the  Mexican  Boundary  Survey  may 
be  given.  At  this  station  TOO  observations  were  made  on  75 
pairs,  25  of  the  pairs  being  observed  twice  each,  and  the 
other  50  once  each.  The  observations  extended  over  four 
nights. 

The  sum  of  the  squares  of  the  fifty  residuals,  A,  obtained 
by  subtracting  the  mean  for  each  pair  which  was  observed 
twice,  from  each  of  the  two  values  from  that  pair,  was  2.50 
square  seconds.  The  probable  error  of  a  single  observation 
was  then,  from  (64), 


The  indiscriminate  mean  of  the  75  results,  one  from  each 
pair,  was  found  to  be  31°  19'  59".  02.  By  subtracting  this 
value  from  each  of  the  75  separate  values,  squaring,  and 
adding,  it  was  found  that  \vv~\  =  11.62  square  seconds. 
From  (65)  it  followed  that 


=  ±  o".267. 


1/4  v      GEODETIC  ASTRONOMY.  §  158. 

The   term    (•  --  1  ---  1  --  .    .    .)  of  formula  (6?)  is  here 
\nl        n^        ns  I 

(25X0.5)+  50=  62.5,  and 


2.5)  =  0.0379.    ,•  ' 

Whence  from  (68) 
(      ^'  =  0.0714—0.0379  =  0.0335     or     es=±o".i$$. 

Substituting  the  values  of  ef  and  e*  in  (69),  the  weights 
for  the  pairs  observed  twice  was  found  to  be  17.8,  and  for 
those  upon  which  one  observation  only  was  made,  12.7. 
Since  it  is  the  relative  weights  only  which  affect  the  final 
result  these  weights  were  for  convenience  written  i.oand  0.7, 
respectively. 

The  resulting  weighted  mean  as  indicated  in  (70)  was 
found  to  be  31°  19'  59".  01. 

From  the  residuals  corresponding  to  this  value  it  was 
found  that  \wv'*~\  —  9.26  square  seconds.  Hence 


Determination  of  Micrometer  and  Level  Values. 

158.  The  most  advantageous  method  of  determining  the 
screw  value  is  to  observe  the  time  required  for  a  close  circum- 
polar  star  near  elongation  to  pass  over  the  angular  interval 
measured  by  the  screw.  Near  elongation  the  apparent  motion 
of  the  star  is  very  nearly  vertical  and  uniform. 

That  one  of  the  four  close  circumpolars  given  in  the 
Ephemeris,  namely,  # ,  #,  and  ^  Ursae  Minoris,  and  5 1  Cephei, 


§158.  MICROMETER    VALUE.  175 

may  be  selected  which  reaches  an  elongation  at  the  most  con- 
venient hour.  In  selecting  the  star  it  may  be  assumed  that 
the  elongations  for  such  stars  occur  when  the  hour-angle  is  six 
hours,  on  either  side  of  the  meridian.  Having  selected  the 
star,  it  is  necessary  both  for  planning  the  observations  and  for 
the  computation  to  compute  the  time  of  elongation  more 
accurately.  The  spherical  triangle  defined  by  the  pole,  the 
star,  and  the  zenith  is  necessarily,  for  a  star  at  elongation, 
right-angled  at  the  star.  Hence  it  may  be  shown  that 

cos  tE  =  tan  0  cot  d,    .          ...     (72) 

in  which  tE  is  the  hour-angle  of  the  star  at  elongation  (always 
less  than  6h).  This  hour-angle  added  to  or  subtracted  from 
the  right  ascension  of  the  star,  for  a  western  or  eastern  elon- 
gation respectively,  gives  the  sidereal  time  of  elongation, 
whence  the  chronometer  time  of  elongation  becomes  known 
by  applying  the  chronometer  error.  It  is  advisable  to  have 
the  middle  of  the  observations  about  at  elongation.  The 
observer  should  obtain  an  approximate  estimate  of  the  rate  at 
which  the  star  moves  along  his  micrometer  by  a  rough  obser- 
vation or  from  previous  record,  and  time  the  beginning  of  his 
observations  accordingly. 

Everything  being  ready  for  the  observations,  the  star  is 
brought  into  the  field  of  the  telescope,  the  telescope  clamped, 
the  star  image  so  placed  as  to  be  approaching  the  micrometer 
line  with  the  micrometer  reading  some  exact  integral  number 
of  turns,  and  the  level  bubble  brought  near  to  the  middle  of 
the  tube.  The  chronometer  time  of  transit  of  the  star  across 
the  line  is  observed,  and  the  level  read.  Then  the  micrometer 
line  is  moved  one  whole  turn  in  the  direction  of  the  motion 
of  the  star,  the  time  of  transit  is  again  observed  and  the  level 
read,  and  the  process  repeated  until  as  much  of  the  middle 
portion  of  the  screw  has  been  covered  by  the  observations  as 


GEODETIC  ASTRONOMY.  §  159. 

is  considered  desirable.  If  desired,  an  observation  may  be 
made  at  every  half-turn,  or  even  every  quarter-turn  by  allow- 
ing an  assistant  to  read  the  level. 

159.  In  Fig.  22,  let  P  be  the  pole,  5  the  position  of  the 
star  when  observed  transiting  across  the  micrometer  line,  and 
SB  its  position  at  elongation.  The  small  circle  SSE  is  a  por- 
tion of  the  apparent  path  of  the  star.  Let  SK  be  a  portion 
of  the  vertical  circle  through  5  limited  by  the  great  arc  PSs* 
Let  the  length  (in  seconds  of  arc)  of  SK,  measuring  the 
change  in  zenith  distance  of  the  star  in  passing  from  the  posi- 
tion 5  to  elongation,  be  called  z.  Let  the  sidereal  interval 
of  time,  in  seconds,  from  position  5  to  elongation  be  called  r. 
In  the  spherical  triangle  SKP  the  angle  at  K  is  90°,  and  that 
at  Pis,  in  seconds  of  arc,  I$T.  Therefore 

sin  z  —  cos  d  sin  (I$T)  .....     (73) 

The  various  values  of  z  corresponding  to  observed  values 
of  r  might  be  directly  computed  from  (73).  But  the  compu- 
tation becomes  much  easier  and  shorter,  though  based  upon 
a  more  difficult  conception,  if  one  proceeds  as  follows: 

(73)  may  be  written 


sn 

(74) 


From  the  known  expansion  of  the  sine  in  terms  of  the  arc 
there  is  obtained 


sin(iSr)  =  (ijr)  sin  i"-i(i5rsin  i")'+T^(i5r  sin  i")'.    (75) 
By  substitution  from  (75)  in  (74) 

*  =  15  cos<5Sr-i(l5  sin  i")V  +  Th<i5  sin  i")Y|.   (76) 


§  160.  MICROMETER    VALUE. 

By  inspection  of  (76)  it  is  evident  that  15  cos  $  is  the  rate 
of  change  of  z  (in  seconds  of  arc  per  second  of  time)  at  elonga- 
tion. If  the  rate  of  change  of  z  were  always  constant  at  this 
value,  z  for  any  position  of  the  star  would  be  (15  cos  tf)r. 
Compare  this  with  (76),  and  it  becomes  evident  that  if  the 
observed  value  of  t  is  corrected  by  applying  to  it  the  small 
correction  —  £(15  sin  i")V  +  ^(15  sin  i")V,  the  resulting 
value  is  that  at  which  the  value  of  z  would  be  the  same  as  the 
actual  value  if  instead  of  the  actual  star  there  were  substituted 
one  whose  motion  is  vertical  at  the  constant  rate  15  cos  8. 
These  corrections  —  \(\  5  sin  i")V  +  TJhrOS  sin  i")4r6  are 
tabulated  in  §  306  for  the  argument  r.  By  using  the  signs 
there  indicated  they  may  be  applied  directly  to  the  observed 
chronometer  times.  The  corrected  times  correspond  to  uni- 
form motion  in  a  vertical  great  circle,  in  the  place  of  the 
actual  motion  in  a  small  circle. 

160.  Suppose  the  reading  of  the  level  has  not  remained 
constant  during  the  observations.  The  change  of  reading 
indicates  that  the  inclination  of  the  telescope  as  a  whole  has 
changed.  Let  #„,  s0  be  the  readings  of  the  north  and  south 
ends  of  the  bubble,  respectively,  at  some  selected  time  during 
the  observations.  Let  it  be  proposed  to  reduce  the  observed 
times  to  what  they  would  have  been  if  the  level  readings  had 
been  «0,  s0  throughout  the  observations.  If  n  and  s  are  the 
north  and  south  readings,  respectively,  at  a  given  observation, 
the  correction  to  the  inclination  of  the  telescope  to  reduce  it 
to  the  state  «0,  s0  of  the  level  is  evidently,  in  seconds  of  arc, 
for  a  graduation  numbered  both  ways  from  the  middle, 

iK« -*)-(«. -OK 

in  which  d  is  the  value  of  one  division  in  seconds  of  arc. 
From  (76),  noting  the  smallness  of  the  tabular  values  of  §  306, 


178  GEODETIC  ASTRONOMY.  §  l6l. 

it  is  evident  that  with  sufficient  accuracy  for  the  present  pur- 
pose the  rate  of  change  of  z  is  15  cos  d  (in  seconds  of  arc 
per  second  of  time)  at  any  instant  during  the  observations. 
Hence  the  correction,  in  seconds,  to  the  observed  chronometer 
time  is 


The  upper  sign  is  to  be  used  for  western  elongation  and  the 
lower  for  eastern.  The  corresponding  formula  for  a  level  with 
a  graduation  numbered  continuously  from  one  end  to  the 
other,  with  the  zero  nearest  the  eyepiece,  is 


(78) 


161.  Having  applied  the  corrections  of  §  306,  and  of 
formula  (77)  or  (78),  to  the  observed  chronometer  times,  the 
results  are  the  times  which  would  have  been  obtained  if  the 
star  had  moved  uniformly  in  a  vertical  circle  and  the  telescope 
had  remained  fixed  in  position.  We  may  now  subtract  the 
first  corrected  time  from  the  middle  one  of  the  series,  the 
second  from  the  next  following  the  middle,  and  so  on,  thus 
obtaining  a  series  of  values  of  the  time  interval  corresponding 
to  a  known  number  of  turns  of  the  micrometer.  The  mean 
of  these  values  may  be  taken,  and  from  it  the  time  interval  t, 
in  seconds,  corresponding  to  one  turn  may  be  derived.  The 
value  of  one  turn,  expressed  in  seconds  of  arc,  is  (15  cos  #)/. 

To  this  must  still  be  applied  corrections  for  differential 
refraction  and  rate  of  chronometer.  The  refraction  correction 
to  the  derived  value  of  one  turn  is  the  change  of  refraction, 
or  differential  refraction,  corresponding  to  one  turn  at  the 


§  162.  MICROMETER    VALUE. 

star's  altitude.  This  is  most  conveniently  derived  by  multi- 
plying the  value  of  one  turn  in  seconds  of  arc  by  one-sixtieth 
of  the  change  of  refraction  for  one  minute  at  the  star's  altitude 
as  given  in  §  294.  The  refraction  correction  may  be  derived 
more  exactly  from  formula  (58),  §  150.  The  correction  is 
always  negative,  as  the  effect  of  refraction  is  always  to  make 
a  heavenly  body  appear  to  move  slower  than  it  actually  does. 
The  correction  for  rate  of  chronometer  may  be  computed 
from  the  proportion: 

rate  of  chronometer  in       required  correction  to  value 
seconds  per  day  of  one  turn 

E • — . 

86  400  value  of  one  turn 

(86  400  seconds  =  I  day.)  The  correction  is  negative  if  the 
chronometer  runs  too  fast,  otherwise  positive.  If  the  value  of 
one  turn  is  approximately  one  minute,  this  correction  is  less 
than  o".ooi  if  the  rate  of  the  chronometer  is  less  than  I3.4 
per  day. 

162.  At  Astronomical  Station  No.  10,  on  the  Mexican 
Boundary  Survey,  the  value  of  the  micrometer  of  the  zenith 
telescope  was  determined  by  observations  upon  Polaris  near 
eastern  elongation  on  October  4,  1892.  The  computed 
chronometer  time  of  elongation  was  2ih  34™  30^.0.  The  rate 
of  the  chronometer  on  sidereal  time  was  o3./  per  day,  gaining. 
The  declination  of  Polaris  was  88°  44'  06". 9.  The  latitude 
of  the  station  was  31°  20'.  The  star  was  observed  at  every 
half-turn  from  the  reading  30  turns  to  the  reading  10.5  turns. 
The  level  numbering  was  continuous  from  one  end  to  the 

other  with  the  zero  nearest  the  eyepiece.   —     - — _  —  IS>94» 

30  cos  d 

Below  is  a  portion  of  the  record  and  computation. 


i8o 


GEODETIC  ASTRONOMY. 


§163. 


Level 

| 

§       c/5 

c 

c 

Readings. 

I 

fl 

g        !T 

(J 

V 

H 

J.f?  . 

Chronometer 

o  rt 

O  «-• 

Corrected 

JM 

Time. 

•i;  c3 

o  > 

^2    z 

u 

Times. 

O  rt  i- 

&&H 

N. 

S. 

13 

us 

Is     6 

i 

II 

£ 

H 

u 

J 

$ 

22.5 

2Ih    20m  51§    0 

5°-4 

I3m  39-  o 

+  o".5 

+  O.2O 

+  0-.4 

2Ih    20m  SI8.  9 

22 

22       23      O 

50.5 

17. 

12       07      0 

+  o  .3 

+  0.10 

+  0   .2 

22      23   .5 

21-5 
21 

23     57    5 
25     32    5 

50-4 
SO.  6 

17- 
17- 

10     32    5 
8     57    5 

+  0  .2 
+  O   .  I 

+  0.30 
+  0.10 

+  o  .6 

+  0.2 

23      58    -3 

25    32  .8 

20.5 

27    05    o 

50.6 

17- 

7     25    o 

+  0   .1 

+  O.IO 

+  O  .2 

27      05    .3 

12.5 

52    08    5 

Si-* 

17- 

'7     38    5 

—  I    .0 

-  0.40 

-o  .8 

52      06  .7 

3im  i4«.8 

12 

II 

53     44    o 
55     16    o 
56     50    o 

51-2 
51-3 
51.2 

17- 

19     14    o 

20      46     0 
22       20     0 

-  1  .3 
-  1  .7 

—    2    .1 

—  0.50 
—  0.60 
—  0.50 

—  1  .0 

—  I    .2 
—  I    .O 

53    4i  -7 
55     13  -i 
56    46  .9 

18  .2 
14.8 
14  .1 

I0.5 

58     25  .0 

51.2 

17- 

23       55    -0 

--    2    .6 

—  0.50 

—  I    .0 

58     21  .4 

16  .1 

Mean  for  10  turns  (from  all  the  observations)  =  31 
Mean  time  for  i  turn  =  187*. 465. 


14". 65  ±  o». 


Log  187.465  =  2.2729202 
Log  cos  d  =  8.3438467 
Log  15  =  1.1760913 


Log  62". 067    =  1.7928582 

=     62".o67 
Correction  for  refraction  *  =  —  o  .051 
"  "    rate  =       o  .000 


One  turn 


Final  value 


=     62  .016  ±  o".on 


163.  If  the  values  of  both  the  level  and  the  micrometer 
are  unknown,  one  may  observe  for  micrometer  value  as  out- 
lined above,  and  also  derive  the  value  of  the  level  in  terms  of 
the  micrometer  as  indicated  in  §  118.  We  may  first  compute 
the  micrometer  value,  omitting  the  level  corrections;  then 
derive  the  value  of  the  level  division  from  this  approximate 
value  of  micrometer.  The  level  corrections  being  now  intro- 
duced into  the  micrometer  computation  will  be  found  to 
modify  it  so  slightly  that  a  second  approximation  for  the  level 
value  will  not  ordinarily  be  required. 

*  Refraction  at  this  station  was  only  |  of  that  at  sea-level. 


§  1  65.  EXROKS.  l8l 

164.  If  no  special  observations  for  micrometer  value  have 
been  made,  or  if  such  observations  have  proved  defective,  the 
micrometer  value  may  be  derived  directly  from  the  latitude 
observations.  Let  <f>P  be  the  mean  latitude,  as  deduced  with 
an  approximate  micrometer  value,  from  all  pairs  for  which  the 
micrometer  difference  (taken  S  —  N]  was  positive,  4>N  the  mean 
latitude  from  pairs  with  minus  micrometer  differences,  DP  the 
mean  of  the  positive  micrometer  differences,  and  DN  the  mean 
of  the  negative  differences.  Then  the  correction  to  the 
approximate  value  of  one  turn  is  * 


For  methods  of  determining  the  level  value  alone,  see  §§ 
II6-I2I. 

Discussion  of  Errors. 

165.  The  external  errors  affecting  a  zenith  telescope 
observation  are  those  due  to  defective  declinations  and  those 
due  to  abnormal  refraction. 

The  declinations  used  in  the  computation  have  probable 
errors  which  are  sufficiently  large  to  furnish  much,  often  more 
than  one-half,  of  the  error  in  the  final  computed  result.  This 
arises  from  the  fact  that  a  good  zenith  telescope  gives  results 
but  little  inferior  in  accuracy  to  those  obtained  with  the  large 
instruments  of  the  fixed  observatories  which  are  used  in 
determining  the  declinations. 

The  following  three  examples  will  serve  to  indicate  the 

*  This  formula  is  not  exact,  from  the  least  square  point  of  view,  that  is, 
it  does  not  give  the  most  probable  value  of  the  required  correction.  But  it 
gives  so  nearly  the  same  numerical  results  as  the  exact  least  square  treat- 
ment, and  leads  to  so  short  and  simple  a  computation,  that  its  use  is  advis- 
able. 


1 82  GEODETIC  ASTRONOMY.  §  1 66. 

improvement  in  the  available  declinations  during  the  last  few 
years,  and  the  magnitude  of  the  declination  errors  to  be 
expected.  The  probable  error  of  the  mean  of  two  declina- 
tions, es,  was  found  *  to  be  ±  0^.5  5  for  the  list  of  stars 
furnished  to  the  U.  S.  Lake  Survey  by  Prof.  T.  H.  Safford 
in  1872.  Similarly,  for  the  list  of  stars  furnished  from  the 
Coast  and  Geodetic  Survey  Office  for  use  in  determining  the 
variation  of  latitude  at  the  Hawaiian  Islands  in  1891-92^ 
es  —  ±  o".i8;  for  the  list  furnished  to  the  Mexican  Boun- 
dary Survey  by  Prof.  Safford  in  1892-93,  es  =  ±o".  18. 
Such  a  high  degree  of  precision  as  that  of  the  last  two  exam- 
ples is  only  attainable  by  an  up-to-date  computation  from 
many  catalogues  of  many  observatories.  By  the  time  such 
lists  are  available  in  print  their  accuracy  has  ordinarily 
diminished  considerably  with  the  lapse  of  time. 

The  errors  in  the  computed  differential  refractions  are 
probably  very  small,  and  it  is  not  likely  that  they  increase 
much  with  an  increase  of  the  mean  zenith  distance  of  a  pair, 
up  to  the  limit,  45°.  If  there  were  a  sensible  tendency,  as 
has  been  claimed,  for  all  stars  to  be  seen  too  far  north,  or 
south,  on  some  nights, — because  of  the  existence  of  a 
barometric  gradient  for  example, — it  should  be  detected  by  a 
comparison  of  the  mean  results  for  different  nights  at  the  same 
station.  Many  such  comparisons  made  by  the  writer  indicate 
that  in  zenith  telescope  latitudes  there  is  no  error  peculiar  to 
the  night.  The  variation  in  the  mean  results  from  night  to 
night  was  found  in  all  the  cases  examined  to  be  about  what 
should  be  expected  from  the  known  probable  errors  of  obser- 
vation and  declination. 

166.  The  observer  s  errors  are  those  made  in  bisecting  the 
star,  and  in  reading  the  level  and  micrometer.  Here  also  may 

*  Professional  Papers  of  the  Corps  of  Engineers,  No.  24,  pp.  622-638. 
f  Coast  and  Geodetic  Survey  Report,  1892,  Part  2,  p.  158. 


§  1 66.  EXROXS.  183 

perhaps  be  classed  the  errors  due  to  unnecessary  longitudinal 
pressure  on  the  head  of  the  micrometer. 

Indirect  evidence  indicates  that  the  error  of  bisection  of 
the  stars  is  one  of  the  largest  errors  concerned  in  the  measure- 
ment. It  probably  constitutes  the  major  part  of  the  computed 
error  of  observation,  and  the  bisection  should  be  made  with 
corresponding  care. 

With  care  in  estimating  tenths  of  divisions  on  the  micro- 
meter head  and  on  the  level  tube,  each  of  these  readings  may 
be  made  with  a  probable  error  of  ±  o.  I  division.  For  the 
ordinary  case  of  a  micrometer  screw  of  which  one  turn  repre- 
sents about  60",  and  of  a  level  of  which  the  value  is  about  i" 
per  division,  such  reading  would  produce  probable  errors  of 
±  o/x.O4  and  ±  0^.05,  respectively,  in  the  latitude  from  a 
single  observation.  These  errors  are  small,  but  by  no  means 
insignificant  when  it  is  considered  that  for  first-class  observing 
the  whole  probable  error  of  a  single  observation,  arising  from 
all  sources  except  declination,  is  less  than  ±  o".3O,  and 
sometimes  even  less  than  ±  o".2O. 

While  reading  the  level  the  observer  should  keep  in  mind 
that  a  very  slight  unequal  or  unnecessary  heating  of  the  level 
tube  may  cause  errors  several  times  as  large  as  the  mere  read- 
ing error  indicated  above;  and  that  if  the  level  bubble  is  found 
to  be  moving,  a  reading  taken  after  allowing  it  to  come  to  rest 
deliberately  may  not  be  pertinent  to  the  purpose  for  which  it 
was  taken.  The  level  readings  are  intended  to  fix  the  posi- 
tion of  the  telescope  at  the  instant  when  the  star  was  bisected. 

It  requires  great  care  in  turning  the  micrometer  head  to 
insure  that  so  little  longitudinal  force  is  applied  to  the  screw 
that  the  bisection  of  the  star  is  not  affected  by  it.  A  dis- 
placement of  3-gVir  part  of  an  inch  in  the  position  of  the 
micrometer  line  relative  to  the  object-glass  produces  in  the 
telescope  of  Fig.  20  a  change  of  more  than  i"  in  the  apparent 


1 84  GEODETIC  ASTRONOMY.  §  1 67. 

position  of  the  star.  The  whole  instrument  being  elastic,  the 
force  required  for  even  such  a  displacement  is  small.  An 
experienced  observer  has  found  that  in  a  series  of  his  latitude 
observations,  during  which  the  level  was  read  both  before  and 
after  the  star  bisections,  the  former  readings  continually 
differed  from  the  latter,  from  o".  I  to  C/'.Q,  always  in  one 
direction.* 

167.  Among  the  instrumental  errors  may  be  mentioned 
those  due:  1st,  to  an  inclination  of  the  micrometer  line  to  the 
horizontal;  2d,  to  an  erroneous  level  value;  $d,  to  inclina- 
tion of  the  horizontal  axis;  4th,  to  erroneous  placing  of  the 
azimuth  stops;  5th,  to  error  of  collimation;  6th,  to  irregu- 
larity of  micrometer  screw;  7th,  to  an  erroneous  mean  value 
of  the  micrometer  screw;  8th,  to  the  instability  of  the  relative 
positions  of  different  parts  of  the  instrument. 

The  first-mentioned  source  of  error  must  be  carefully 
guarded  against,  as  indicated  in  §  139,  as  it  tends  to  introduce 
a  constant  error.  The  observer,  even  if  he  attempts  to  make 
the  bisection  in  the  middle  of  the  field  (horizontally),  is  apt  to 
make  it  on  one  side  or  the  other  according  to  a  fixed  habit. 
If  the  line  is  inclined  his  micrometer  readings  are  too  great  on 
all  north  stars  and  too  small  on  all  south  stars,  or  vice  versa. 

The  error  from  using  an  erroneous  level  value  is  smaller 
the  smaller  are  the  level  corrections  and  the  more  nearly  the 
plus  and  minus  corrections  in  a  series  balance  each  other.  To 
insure  that  it  shall  be  negligible  it  is  necessary  to  relevel  every 
time  the  correction  becomes  more  than  two  seconds,  at  most. 

The  errors  from  the  third,  fourth,  and  fifth  sources  may 
easily  be  kept  negligible.  An  inclination  of  one  minute  of  arc 
in  the  horizontal  axis,  or  an  error  of  that  amount  in  either 
collimation  or  azimuth,  produces  only  about  o".oi  error  in  the 

*  Coast  and  Geodetic  Survey  Report,  1892,  Part  2,  p.  58. 


§  1 68.  EXXOXS.  185 

latitudes.  All  three  of  these  adjustments  may  easily  be  kept 
far  within  this  limit. 

Most  micrometer  screws  are  so  regular  that  the  unelimi- 
nated  error  in  the  mean  result  for  a  station  from  the  sixth 
cause  is  usually  very  small.  But  it  should  not  be  taken  for 
granted  that  a  given  screw  is  regular.  Large  irregularities 
may  be  detected  by  inspection  of  the  computation  of  the 
micrometer  value.  Errors  with  a  period  of  one  turn  may  be 
detected  by  making  the  observations  for  micrometer  value  at 
every  quarter-turn,  and  then  deriving  the  value  of  each 
quarter,  o  to  25  divisions,  25  to  50  divisions,  etc.,  of  the  head 
separately.  The  four  mean  values  thus  derived  should  agree 
within  the  limits  indicated  by  their  probable  errors. 

168.  To  guard  against  error  from  the  seventh  source  the 
pairs  must  be  so  selected  as  to  make  the  plus  and  minus 
micrometer  differences  at  a  station  balance  as  nearly  as  possi- 
ble,* For  example,  at  the  fifteen  astronomical  stations 
occupied  on  the  Mexican  Boundary  Survey  of  1892-93  the 
mean  micrometer  difference,  taken  with  regard  to  sign,  never 
exceeded  0.36  turn  at  any  station,  and  was  less  than  o.  10  turn 
at  nine  of  the  stations.  If  the  plus  and  minus  micrometer 
differences  balance  exactly  at  a  station,  an  erroneous  microm- 
eter value  does  not  affect  the  computed  latitude,  but  merely 
increases  the  computed  probable  errors. 

*  It  seems  an  easy  matter  to  make  an  accurate  determination  of  mi- 
crometer value.  But  experience  shows  that  such  determinations  are 
subject  to  unexpectedly  large  and  unexplained  errors.  For  example,  in 
the  Hawaiian  Island  series  of  observations,  mentioned  above,  the  microm- 
eter value  was  carefully  determined  twelve  times.  The  results  show  a 
range  of  nearly  ^  of  the  total  value.  This  corresponds  to  a  range  of 
about  one-sixth  of  an  inch  in  the  focus  of  the  object-glass.  In  the  San 
Francisco  series,  and  in  general  wherever  the  micrometer  value  has  been 
repeatedly  measured,  the  same  large  discrepancies  have  been  encoun- 
tered. Hence  the  need  of  carrying  out  the  suggestions  of  the  above  para- 
graph. 


1 86  GEODETIC  ASTRONOMY.  §  169. 

The  errors  from  the  eighth  source  may  be  small  on  an 
average,  but  they  undoubtedly  produce  at  times  some  of  the 
largest  residuals.  They  may  be  guarded  against  by  protect- 
ing the  instrument  from  sudden  temperature  changes,  and 
from  shocks  and  careless  handling,  and  by  avoiding  long  waits 
between  the  two  stars  of  a  pair.  The  closer  the  agreement 
in  temperature  between  the  instrument  room  and  the  outer 
air  the  more  secure  is  the  instrument  against  sudden  and 
unequal  changes  of  tempejature. 

The  computed  probable  error  of  a  single  observation,  e, 
including  all  errors  except  those  of  declination,  was  found  to 
be  as  follows  in  three  recent  first-class  latitude  series:  In  the 
observations  for  variation  of  latitude  at  San  Francisco  *  in 
1891-92,  1277  observations  (in  two  series)  gave  e  =  ±  o".  19 
and  e  =  ±  o".28  ;  in  a  similar  series  at  the  Hawaiian  Islands  f 
for  the  same  purpose  in  1891-92,  2434  observations  gave 
e=  ±o".i6;  from  1362  observations  at  fifteen  stations  on 
the  Mexican  Boundary  in  1892-93,  e  =  ±  o" '.19  to  ±  O7/.38.J 

169.  When  an  observer  begins  planning  a  series  of  obser- 
vations to  determine  the  latitude  of  a  given  point  two  ques- 
tions at  once  arise.  How  many  observations  shall  be  made  ? 
How  many  separate  pairs  shall  be  observed  ?  Increasing  the 
number  of  observations  increases  the  cost  of  both  field  work 
and  computation.  An  increase  in  the  total  number  of 
separate  pairs  adds  proportionally  to  the  work  of  computing 
the  mean  places,  but  otherwise  has  little  effect  on  the 
total  cost.  The  economics  of  the  problem  demand  that  the 
ratio  of  observations  to  pairs  shall  be  such  as  to  give  the 
maximum  accuracy  for  a  given  expenditure.  Two  extremes 

*  Coast  and  Geodetic  Survey  Report,  1893,  Part  2,  p.  494. 
f  Coast  and  Geodetic  Survey  Report,  1892,  Part  2,  pp.  54,  158. 
\  Transactions  of  the  Association  of  Civil  Engineers  of  Cornell  Uni- 
versity, 1894,  p.  58. 


§  170.  ERRORS.  187 

of  practice  are  to  take  210  observations  on  30  pairs,  each  pair 
being  observed  on  7  nights;  and  to  take  100  observations  on 
100  pairs,  each  pair  being  observed  but  once.  The  first  is 
the  old  practice  of  the  Coast  Survey.*  The  recent  practice 
of  that  Survey  is  intermediate  between  these  extremes.  The 
latter  extreme  was  approached,  but  not  quite  reached,  on  the 
Mexican  Boundary  Survey  of  1892-93. 

Let  it  be  supposed  that  e  —  ±  o".2i  and  e$  =  ±  o".  18, 
as  in  the  example  given  in  §  157.  Then  for  the  former 
extreme  method  the  effect  of  the  errors  of  observation  on  the 
result  would  be  reduced  to  ±  o".2i  -r-  1/2 10  —  ±  o".oi4, 
and  the  effect  of  the  declination  errors  to  ±  o".  18  ~  V$o 
=  ±  Q^.033,  giving  for  e^  the  probable  error  of  the  result, 

4/(o.oi4)2  +  (0.033)'  =  ±  o".036.  In  the  latter  extreme 
case  the  error  of  observation  would  be  reduced  to  ±  o".2i 
-r-  Vioo  =  ±o//.O2i,  the  declination  error  to  ±o".i8 
-f-  ^100  =  it  o".oi8,  and  the  probable  error  of  the  result  to 

4/(o.02i)2  +  (0.018)'  =  ±  o".028.  To  look  at  the  matter  in 
another  light :  if  with  the  above  data  as  to  e  and  es  the  weight 
for  a  pair  observed  once  is  called  unity,  that  for  a  pair 
observed  twice  is  1.40,  by  formula  (69),  §  156;  observed 
seven  times  is  1.98;  and  for  a  pair  observed  an  infinite 
number  of  times  2.36.  Little  is  gained  in  accuracy  from  the 
second  observation  on  a  pair,  and  less  from  each  succeeding 
observation. 

170.  The  zenith  telescope  furnishes  a  latitude  determina- 
tion which  is  so  far  superior  to  that  given  by  any  other  portable 
instrument  that  it  should  always  be  used  where  great  accuracy 
is  desired.  A  theodolite,  or  an  astronomical  transit,  may  be 
used  as  a  zenith  telescope  if  furnished  with  a  suitable  eyepiece 
micrometer,  and  with  a  sufficiently  sensitive  level  parallel 

*  C.  &  G.  S.  Report,  1893,  Part  2,  p.  301. 


188  GEODETIC  ASTRONOMY.  §  I/I. 

to  the  plane  in  which  the  telescope  rotates  upon  its  horizontal 
axis.  For  the  convenience,  however,  of  those  who  may 
desire  to  determine  the  latitude  with  a  sextant  on  explora- 
tions or  at  sea,  and  of  those  who  may  be  forced  by  circum- 
stances to  determine  the  latitude  by  a  measurement  of  the 
altitude  of  the  Sun,  or  a  star,  with  a  theodolite  or  an  alt- 
azimuth, the  following  formulae  are  here  collected: 

To  compute  the  latitude  from  an  observed  altitude  of  a  star, 
or  the  Sun,  in  any  position,  the  time  being  known. 

171.  The  requisite  formulae  are 

tan  D  =  tan  tf  sec  /, (81) 

cos  (0  —  D)  =  sin  A  sin  D  cosec  6 ;      .     .     (82) 

in  which  6  is  the  declination  and  /  the  hour-angle  of  the  star 
(or  Sun)  at  the  instant  of  observation;  D  is  an  auxiliary 
angle  introduced  merely  to  simplify  the  computation ;  A  is 
•the  altitude  resulting  from  the  measurement  after  applying  all 
instrumental  corrections,  the  correction  for  refraction,  and, 
if  the  Sun  is  observed,  the  corrections  for  parallax  and  semi- 
diameter  (see  §§  65,  66).  D  is  to  be  taken  less  than  90°,  and 
-|-  or  —  according  to  the  algebraic  sign  of  the  tangent. 
Formula  (82)  is  ambiguous  in  that  0  —  Z>,  determined  from 
the  cosine,  may  be  either  positive  or  negative.  But  the 
latitude  of  the  station  is  always  known  beforehand  with  suffi- 
cient accuracy  to  decide  between  these  two  values.  These 
formulae  are  exact,  no  approximations  having  been  made  in 
deriving  them.* 

*  For  this  derivation  see  Doolittle's  Practical  Astronomy,  pp.  236,  237; 
or  Chauvenet's  Astronomy,  vol.  I.  pp.  229,  230. 


§  172.  LATITUDE  FROM  ZENITH  DISTANCES.  189 

To  compute  the  latitude  from  zenith  distances  of  a  star,  or 
the  Sun,  observed  near  the  meridian,  the  time  being  known. 

172.  The  rate  of  change  of  zenith  distance  (or  of  altitude) 
of  a  given  star  is  smaller  the  nearer  the  star  is  to  the  meridian. 
Hence  the  effect  of  a  small  error  in  the  time,  which  is 
assumed  to  be  known,  is  less  the  nearer  the  observation  is 
made  to  the  meridian,  and  is  zero  for  an  observation  made 
precisely  on  the  meridian.  Only  a  single  pointing  can  be 
made  when  the  star  is  on  the  meridian,  whereas  it  is  desirable 
to  take  several  pointings  so  as  to  decrease  the  effect  of  errors 
of  observation.  •  Hence  the  desirability  of  a  rapid  method  for 
computing  the  latitude  from  circummeridian  observations. 

Let  0  be  the  required  latitude  of  the  station;  6  the 
declination  of  the  star  at  observation;  £,,  Cf,  C3,  .  .  .  succes- 
sive observed  values  of  the  zenith  distance  of  the  star  corre- 
sponding to  the  hour-angles  t^  t^  A,,  .  .  .  ;  and  C0  the 
meridional  zenith  distance  of  the  star.  Then 


.  (83) 


2  sin'  \t  2  sin6  \t 


0  = 


sin 


77-.      .        (84) 

i" 


A,  B,  and  C  are  evidently  constant  for  a  series  of  obser- 
vations made  near  a  given  meridional  passage  of  the  star. 
Let  mlt  mv  m»  .  .  .  ,  nlt  «a,  *,,'...,  <?„  0a,  <?„...,  be 
values  of  m,  n,  and  o  corresponding,  respectively,  to  /,,  /,, 
/3,  .  .  .  .  Then  for  a  star  upon  which  u  observations  are 
made  near  upper  culmination 


u 

in  which  the  upper  signs  are  to  be  used  if  the  star  crosses  the 


190  GEODETIC  ASTRONOMY.  §  1/2. 

meridian  south  of  the  zenith,  and  the  lower  signs  if  it  crosses 
north  of  the  zenith. 

Similarly,  for  a  star  upon  which  u  observations  are  made 
near  lower  culmination 


/  being  now  reckoned  from  lower  culmination. 

These  formulae  *  are  not  exact.  They  are  derived  by  an 
expansion  into  a  converging  series,  of  which  only  the  first 
three  terms  are  retained.  The  errors  of  the  formulae  are 
greater  the  greater  the  hour-angle,  and  the  smaller  the  zenith 
distance  of  the  star.  Their  accuracy  is  sufficient  for  the 
reduction  of  sextant  observations  if  the  hour-angle  is  limited 
to  30™,  and  is  also  not  allowed  to  exceed  (in  minutes  of  time) 
the  zenith  distance  of  the  star  (in  degrees).  To  be  certain  of 
sufficient  accuracy  for  the  reduction  of  observations  made 
with  a  theodolite  or  altazimuth  the  hour-angle  must  be  limited 
to  3<Dm,  and  to  one-half  (in  minutes  of  time)  the  zenith  dis- 
tance of  the  star  (in  degrees). 

It  will  be  noted  that  0  and  C0  are  required  at  the  start  for 
the  reduction  since  they  occur  in  the  second  member  of  (85) 
and  (86).  Only  approximate  values  are  required,  however, 
as  the  terms  involving  A,  B,  and  C  are  always  small.  Such 
approximate  values  may  be  derived  from  previous  computa- 
tions; from  a  preliminary  approximate  reduction  of  a  single 
observation;  or,  if  the  observations  extend  on  both  sides  of 
the  meridian,  the  smallest  observed  zenith  distance  may  be 
assumed  for  this  purpose  to  have  been  made  upon  the 
meridian. 

The  values  of  m,  n,  and  o  will  be  found  tabulated  in  terms 
of  *  in  §  307,  for  values  of  /  up  to  3Om. 

*  For  their  derivation  see  Doolittle's  Practical  Astronomy,  pp.  238-244; 
or  Chauvenet's  Astronomy,  vol.  i.  pp.  238-240. 


§  173-         LATITUDE  FROM  OBSERVED   ALTITUDE.  1<}1 

If  the  Sun  is  observed,  6  is  not  a  constant  during  the 
period  of  observation.  But  formula  (85)  may  still  be  used  for 
the  reduction  of  sextant  observations  if  for  d  is  substituted 
the  mean  of  the  declinations  corresponding  to  each  instant  of 
observation. 

To  compute  the  latitude  from  observations  of  the  altitude 
of  Polaris  at  any  hour  -angle,  the  time  being  known. 

173.  The  apparent  motion  of  Polaris  is  so  slow  thai;  it 
may  be  observed  at  any  hour-angle  with  the  assurance  that 
the  effect  of  an  error  in  the  assumed  chronometer  correction 
upon  the  computed  latitude  will  be  small.  It  is  so  bright  as 
to  be  quickly  found  and  brought  into  the  field  of  a  telescope. 
For  stations  in  the  United  States  it  is  always  in  a  convenient 
position  for  observation.  These  considerations  often  lead  to 
the  adoption  of  Polaris  as  the  object  when  the  latitude  is  to 
be  determined  by  direct  observations  of  altitude  with  a 
theodolite  or  smaller  instrument. 

If  A  is  the  altitude  of  Polaris  at  the  hour-angle  /,  and  P 
is  its  polar  distance  (=  90°  —  #)  expressed  in  seconds  of  arc, 
then* 


0  =  A  —  P  cos  t  +  %P*  sin  i"  sin2  /  tan  A 

—  fP3  sin2  \"  cos  /  sin2  /  +  fP  sins  i"  sin4  /  tan3  A 

sin3  i"  (4  —  9  sin1  /)  sin2  /  tan  A,      .      .      .     .     (87) 


in  which  all  terms  in  the  second  member  except  the  first  are 
in  seconds  of  arc. 

The  last  term  will  never  exceed  o".oi  at  any  station  of 
which  the  latitude  is  not  greater  than  82°.  The  fifth  term 
will  never  exceed  o".oi  for  any  station  below  latitude  48°, 
nor  o".io  at  any  station  below  67°.  The  maximum  value  of 

*  For  the  derivation  of  this  formula  see  Doolittle's  Practical  Astronomy, 
pp.  256-259;  or  Chauvenet's  Astronomy,  vol.  i.  pp.  253-255. 


I Q2  GEODETIC  ASTRONOMY.  §  1/4. 

the  fourth  term  is  about  Q^.33.  The  preceding  statements 
are  based  upon  i°  20'  as  the  value  of  /*,  and  the  terms  are 
correspondingly  reduced  when  P  is  below  that  limit. 

174.  Formulae  (81),  (82),  and  (87)  furnish  the  means  of 
computing  the  latitude  from  a  single  observation  of  altitude. 
How  must  one  proceed  if  a  series  of  observations  of  the  alti- 
tude have  been  made  ?  If  the  chosen  formula  is  applied  to 
each  observation  separately,  each  computation  will  be  exact ; 
the  computations  will  be  so  nearly  alike  as  to  furnish  con- 
venient rough  checks  by  differences;  but  if  there  are  many 
observations  the  computation  of  the  series  becomes  tediously 
long.  If  one  takes  the  mean  of  all  the  measured  altitudes, 
assumes  that  it  corresponds  to  the  mean  of  the  hour-angles, 
and  applies  either  formula  to  this  mean  altitude  and  mean 
hour-angle,  the  result  is  approximate.  For,  the  mean  of  the 
altitudes  does  not  correspond  to  the  mean  of  the  hour-angles, 
in  general,  because  of  the  curvature  of  the  apparent  path  of 
the  star  and  the  corresponding  variation  in  the  rate  of  change 
of  the  altitude.  The  error  involved  in  the  assumption  that 
the  two  means  correspond,  evidently  decreases  as  the  interval 
covered  by  the  series  of  observations  decreases.  If,  then, 
one  breaks  up  the  series  into  sufficiently  short  groups,  the 
means  for  each  group  may  be  treated  by  a  single  application 
of  the  formula  without  sensible  error.  For  sextant  observa- 
tions it  suffices  when  observing  upon  Polaris  to  break  up  the 
series  into  groups  not  more  than  fifteen  minutes  long.  For 
more  accurate  observations,  made  upon  Polaris  with  a  theodo- 
lite or  altazimuth,  one  cannot  be  certain  of  sufficient  accuracy 
unless  each  group  is  limited  to  about  five  minutes.  In  using 
(81)  and  (82)  with  the  Sun,  or  with  other  stars  than  circum- 
polars,  it  will  usually  be  necessary  to  make  the  groups  very 
much  shorter  than  indicated  above. 


§  175-         LATITUDE  FROM  OBSERVED   ALTITUDE.  193 

175.  The  process  of  making  a  direct  measurement  of  alti- 
tude with  an  instrument  having  a  vertical  circle  (theodolite  or 
altazimuth)  is  so  simple  that  a  detailed  statement  of  it  is 
hardly  necessary  here.  Suffice  it  to  say  that  the  effect  of  the 
index  error  of  the  vertical  circle  must  be  eliminated  from 
each  observation  or  group  of  observations  before  applying  the 
formula  for  the  latitude  computation ;  that  the  chronometer 
time  of  each  pointing  upon  the  star  must  be  noted  with  a 
chronometer  of  which  the  error  is  known  (to  fix  the  hour- 
angle);  and  that  the  inclination  of  the  vertical  axis  must  be 
determined  during  the  observations  by  readings  of  a  level 
having  its  tube  parallel  to  the  plane  of  the  telescope.  The 
effect  of  index  error  may  be  eliminated  either  by  making 
special  observations  to  determine  the  index  error,  by  making 
half  of  the  observations  of  each  group  with  the  telescope  in 
a  reversed  position,  or  by  a  combination  of  both  these  pro- 
cesses. The  accuracy  with  which  the  chronometer  error  must 
be  known  may  be  inferred  from  the  observed  rate  of  change 
of  the  altitude.  The  correction  to  the  measured  altitude  due 
to  the  inclination  of  the  vertical  axis  may  be  computed  by 
formula  (55)  or  (56),  §  148,  using  the  algebraic  signs  there 
given  if  the  star  is  north  of  the  zenith,  and  reversing  them  if 
south  of  the  zenith. 

The  method  of  observing  the  altitude  of  the  Sun,  or  a 
star,  with  a  sextant,  has  already  been  given  in  Chapter  III, 
in  connection  with  the  determination  of  time. 

If  for  a  given  purpose  it  is  only  required  to  determine  the 
latitude  within  30"  the  table  given  in  §  308  may  be  utilized 
as  there  indicated. 


IQ4  GEODETIC  ASTRONOMY. 


QUESTIONS    AND    EXAMPLES. 

176.  I.  State  in  detail  how  you  would  make  the  com- 
puted allowance  for  parallax  upon  the  horizontal  circle  if  a 
single  point  at  a  known  distance  is  used  in  making  the  colli- 
mation  adjustment  of  a  zenith  telescope  mounted  on  one  side 
of  its  vertical  axis.  (See  §  137.) 

2.  Prove  formulae  (55)  and  (56),  for  the  level  corrections 
to  be  applied  to  observations  with  a  zenith  telescope. 

3.  At  the  same  station  and  with  the  same  instrument  as 
given  in  §  152,  the  following  latitude  observations  were  also 
made.     Compute  the  results. 


Date,  1892. 

B.  A.  C. 

No.  of 

N.  or  S. 

Micrometer 
Reading. 

Level 
Readings. 

Declinations. 

Star. 

/ 

d 

N. 

S. 

August 

9 

7380 

S 

10 

70.9 

18.1 

52. 

8 

4° 

48' 

12".  76 

7417 

N 

28 

36.0 

50.8 

16. 

0 

58 

10 

03  .10 

August 

9 

7440 

S 

29 

06.4 

16.9 

51. 

7 

-4 

01 

03    -43 

7482 

N 

9 

09.0 

53-1 

18. 

i 

66 

20 

18  .63 

August 

16 

7440 

S 

30 

21.  1 

12.9 

56. 

5 

-4 

01 

02    .75 

7482 

N 

10 

25-9 

57-6 

13- 

8 

66 

2O 

21    .26 

Diminish  the  differential  refractions  of  §  304  by  2ofo  (see 
§  150).          Ans.   0=  31°  19'  $8".4i,  58".83,  and  59".62. 

4.  What  is  the  correction  to  the  computed  latitude  due 
to  observing  the  star  off  the  meridian,  with  a  zenith  telescope, 
in  each  of  the  following  cases  ?     The  declinations  of  the  four 
stars  were,  respectively,  23°   19',  —  5°  42',  46°  06',   81°  27', 
and  they  were  observed  too  late  by  the  following  intervals, 
respectively,  12s,  2i8,  18%  53s. 

Ans.   +  o".oi,  —  o".oi,  +  o".05,  and+o".n. 

5.  At    Astronomical    Station     No.    5,    on    the    Mexican 
Boundary,  99  observations  with  a  zenith  telescope  were  taken 
on  50  pairs,  each  pair  excepting  one  being  observed  twice. 
It  was  found  that  [JJ]  =  5.17  square  seconds,  and  [vv]  = 


§  1 7 6.  QUESTIONS  AND   EXAMPLES.  195 

7.52  square  seconds.      Compute  e  and  e$.      For  notation  and 
formulae,  see  §§  154,  155. 

Ans.   e  =  ±  o".22  ;  e&  =  ±  o".2i. 

6.  Prove  formula  (72),  for  the  computation  of  the  hour- 
angle    of  a  star  at    elongation.     If   d  is  less  than   0,    this 
formula  leads  to  an  absurdity,  namely,  a  cosine  greater  than 
unity.     To  what  does  this  absurdity  correspond  in  nature  ? 

7.  What  was  the  sidereal  time  of  western  elongation  of 
d  Ursae  Minoris  at  Anchorage  Point,  Chilkat  Inlet,   Alaska 
(0  =  59°  10'   19". 4)  on  October  13,  1892  ?     Its  declination 
was  then  86°  36'  54".!,  and  its  right  ascension  i8h  o6m  498.6. 

Ans.   23h  44m  O48.5. 

8.  State  the  reason  for  the  double  signs  in  formulae  (77) 
and  (78),  for  the  level  corrections  in  a  computation  of  microm- 
eter value,  and  prove  that  the  signs  as  given  are  correct. 

9.  What  is  the  mean  value  of  one  turn  of  the  micrometer 
deduced  from  the  following  observations,  forming  a  portion 
of  the  series  from  which  the  observations  given  in  §  162  were 
also  extracted  ? 


ter  Readir 

t 

ig.    Chronometer  Time. 

Level  Readings. 

N.              S 
d              d 

27.0 

2Ih   O6m 

43'-  5 

50.1 

17.4 

26.5 

08 

18.5 

50.3 

17.7 

26.O 

09 

50.5 

50.2 

17.2 

25-5 
17.0 

II 

27.0 

50.1 
50.9 

17.1 
17.1 

21      38 

03  .0 

I6.5 

39 

35-5 

50.9 

17.1 

16.0 

4i 

12.0 

51.0 

17.2 

15-5 

42 

41-5 

51.0 

17.1 

Ans.  61". 987. 

10.  The  following  altitudes,  uncorrected  for  refraction,  of 
the  star  Altair  (a  =  I9h  45m  323.5,  <?  =  8°  35'  08".  7)  were 
observed  with  a  theodolite  on  October  13,  1892:  51°  53'  28", 
51°  35'  49",  51°  06'  40",  and  50°  48'  09".  The  times  of  the 


I$6  GEODETIC  ASTRONOMY.  §  1/6. 

separate  observations  as  read  from  the  face  of  a  chronometer 
known  to  be  13s.  i  fast  of  local  sidereal  time  were,  respec- 
tively, 2ih  32ra  i68.i,  2ih  34m  I98.6,  2ih  37m  428.5,  and  2ih 
39m  5Oa.  i.  Compute  the  latitude  of  the  station.  Carry  your 
computation  to  tenths  of  seconds  of  arc,  and  treat  each  obser- 
vation separately. 

Ans.   Mean  value  of  0  =  38°  14'  ii". 

11.  The  following  observations  of  the  altitude  of  the  Sun 
were  made  with  a  theodolite  on  February   15,    1892.     The 
times  are  given  as  read  from  the  chronometer,   which  was 
known  to  be   n8  fast  of  local  mean  solar  time,  and  the  alti- 
tudes are  given  after  all  corrections  have  been  applied.     The 
following  data  in  regard  to  the  Sun  were  derived  from  the 
Ephemeris: 

Times.  Corrected  Altitudes. 

IIh    42™  19'  42°    05'     II"  d  =  —  12°  40'  39". 

43  12  05     16  Equation  of  time  =  -f~  i4m  20*. 

44  03  05     27  These  values  correspond  to  the 

44  50  05  29  mean  of  the  observed  times. 

45  41  05  35  What  was  the  latitude  of  the 

46  32  05  30  station  ? 

47  26  05  36 

48  13  05  28 

49  08  05  16 

50  02  04  55 

Ans.   0  =  35°  13' 46". 

12.  Four  observations  of  the  altitude  of  Polaris  with  an 
altazimuth  gave  for  its  mean  altitude,  corrected  for  refraction, 
40°  19'  48". 9.     The  sidereal  times  of  the  observations  were, 
respectively,  22h   19™   i68,  22h  2Om  3 Is,  22h  22m  51%  and  22h 
24™  io8.     At  the  time  of  observation  the  right  ascension  of 
Polaris  was   ih  20™  05". i,  and  its  declination  88°  44'  05". 9. 
What  was  the  latitude  of  the  station? 

Ans.   39°  26'  06".  8. 


§  1 73.  THE  INSTRUMENT.  197 


CHAPTER   VI. 
AZIMUTH. 

177.  The  instrumental  process  of  determining  the  azimuth 
of  a  terrestrial  line  astronomically  consists  of  a  measurement 
of  the  angle  between  two  vertical  planes — one  defined  by  the 
azimuth  mark  and  the  vertical  line  through  the  instrument, 
and  the  other  by  the  observed  star  and  the  vertical  at  the 
instrument.     Since  the  angle   between   these   two  planes  is 
continually  changing,  the  exact  time  at  which  each  pointing 
is  made  upon  the  star  must  be  noted  upon  a  chronometer  of 
which   the   error  is  known.     From   this   recorded   time   the 
hour-angle  of  the  star  and  its  azimuth  as  seen  from  the  station 
may  be  computed.     The  computed  azimuth  of  the  star  com- 
bined by  addition  or  subtraction  (as  the  case  may  be)  with 
the  measured  horizontal  angle  at  the  station  between  the  star 
and  the  azimuth  mark  gives  the  azimuth  of  the  mark  from 
the  station. 

Description  of  Instrument. 

178.  Any    one    of    the    many    theodolites    used    for   the 
measurement  of  the  horizontal  angles  of  a  triangulation  may 
be  used  for  azimuth  determinations, — provided  the  telescope 
can  be  inclined  enough  to  point  to  the  star, — each  instrument 
giving  a  degree  of  accuracy  dependent  upon  its  size,  power, 
and    workmanship.       The    larger   instruments    often    called 
altazimuths,  designed  primarily  for  astronomical  work,  do  not 
differ  in  principle  from  the  smaller  theodolites.     The  instru- 


198  GEODETIC  ASTRONOMY.  §  178. 

ment  shown  in  Fig.  24  is  a  20 -in.*  theodolite  which  belongs 
to  the  U.  S.  Coast  and  Geodetic  Survey. 

The  Troughton  and  Simms  altazimuth  now  in  use  in  the 
College  of  Civil  Engineering  of  Cornell  University  may  be 
taken  as  a  type  of  the  larger  altazimuths.  Its  horizontal 
circle  is  36  cm.  (=  14  in.)  in  diameter,  and  is  graduated  to  5' 
spaces.  It  may  be  quickly  undamped  from  the  fixed  center 
of  the  instrument,  and  its  zero  shifted  to  a  new  position. 
The  index  microscope  carries  in  its  field  of  view  (as  the  comb 
is  carried  in  the  micrometer  of  the  zenith  telescope;  see  §  135) 
a  small  pointer  which  is  seen  projected  against  a  portion  of  the 
horizontal  circle,  and  gives  the  degrees  and  the  nearest  pre- 
ceding five  minutes  of  the  reading.  The  remaining  minutes 
and  the  seconds  of  the  reading  are  obtained  from  three 
reading  microscopes.  Each  of  these  microscopes  is  furnished 
with  an  eyepiece  micrometer  similar  to  that  of  the  zenith 
telescope  described  in  §  135.  A  line  of  the  circle  graduation, 
the  object-glass,  and  the  micrometer  line  bear  the  same 
relation  to  each  other  here  as  the  star,  the  object-glass,  and 
the  micrometer  line  bear  to  each  other  in  the  zenith  telescope 
(see  Fig.  21,  and  the  corresponding  text,  §  135).  Each 
microscope  is  so  adjusted  that  five  turns  of  the  micrometer 
screw  correspond  as  nearly  as  may  be  to  one  space  on  the 
circle.  Each  turn  represents  therefore  approximately  one 
minute,  and  each  of  the  sixty  equal  divisions  of  the  micrometer 
head  one  second.  The  reading  of  the  micrometer  increases 
as  the  micrometer  line  apparently  moves  in  the  direction  of 
decreasing  graduations.  The  reading  given  directly  by  the 
comb  and  head  is  the  distance,  measured  in  the  backward 
direction  along  the  circle  graduation,  from  the  zero  of  the 
micrometer  (middle  of  the  comb)  as  seen  projected  upon  the 
circle,  to  the  nearest  graduation  representing  too  small  a 

*  That  is,  a  theodolite  with  a  horizontal  circle  twenty  inches  in  diam- 
eter. 


§  179-  THE  INSTRUMENT.  1 99 

reading  of  the  circle.  This  reading  added  to  that  of 
the  index  microscope  gives  the  complete  reading  of  the 
circle.  The  three  reading  microscopes  give  nominally 
identical  readings,  and  the  mean  is  taken  to  secure  in- 
creased accuracy.  The  use  of  two  or  more  microscopes 
or  verniers  on  a  circle  also  serves  of  course  to  eliminate, 
wholly  or  in  part,  the  errors  which  would  otherwise  enter 
the  measurements  from  the  effect  of  eccentricity  and  of 
periodic  errors  in  the  graduation  of  the  circle.  Instead  of  a 
single  micrometer  line  with  which  to  bisect  the  graduations, 
these  reading  microscopes  are  usually  provided  with  two 
parallel  lines  at  such  a  distance  apart  that  when  they  are 
placed  symmetrically  on  the  two  sides  of  a  graduation  a 
narrow  strip  of  light  is  seen  between  each  line  and  the  edge 
of  the  dark  graduation.  More  accurate  pointings  upon  a 
graduation  can  be  made  with  such  a  double  line  than  with  a 
single  line. 

The  telescope  has  a  focal  length  of  24  in.,  its  object-glass 
has  a  clear  diameter  of  2j  in.,  and  its  eyepiece  a  magnifying 
power  of  about  45  diameters.  The  value  of  one  division  of 
the  striding  level  is  i".8.  The  vertical  circle  is  graduated  to 
5'  spaces,  and  is  read  to  seconds  by  two  micrometer  micro- 
scopes. The  telescope  is  furnished,  in  addition  to  a  fixed 
reticle,  with  an  eyepiece  micrometer  similar  to  that  of  the 
zenith  telescope,  but  turned  90°  in  its  position  so  that  it 
measures  small  angles  in  the  plane  defined  by  the  telescope 
and  its  horizontal  axis  instead  of  differences  of  zenith  dis- 
tances. (For  the  use  of  this  micrometer  see  §§  205-214.) 
The  other  features  of  the  instrument  are  sufficiently  shown  by 
the  figure. 

179.  The  three  reading  microscopes  are  often,  especially 
on  smaller  instruments,  replaced  by  two  microscopes  or  by 
two  verniers.  A  hanging  level  may  be  used  instead  of  a 
striding  level.  The  eyepiece  micrometer  of  the  telescope,  the 


200  GEODETIC  ASTRONOMY.  §  l8o. 

separate  index  microscope,  and  the  vertical  circle  are  often 
omitted.  A  reflecting  prism  is  sometimes  placed  at  the 
intersection  of  the  telescope  with  the  horizontal  axis  to  turn 
the  light  rays  at  right  angles,  and  the  eyepiece  is  then  placed 
at  the  end  of  the  horizontal  axis.  The  proportions  of  the 
various  parts  and  their  absolute  size  may  be  varied  greatly. 
The  graduated  circle  may  be  furnished  with  a  clamp  and  tan- 
gent screw,  similar  in  principle  and  purpose  to  that  of  the 
lower  motion  of  an  engineer's  transit.  These  various  modifi- 
cations produce  great  changes  in  the  outward  appearance  of 
the  instrument,  may  introduce  certain  obvious  limitations  to 
their  use,  but  the  principles  involved  are  changed  only  in 
minor  details. 

Adjustments. 

180.  The  vertical  axis  must  be  made  truly  vertical  by  the 
same  process  that  is  used  in  levelling  up  an  engineer's  transit. 
Whatever  levels  are  to  be  used  should  for  convenience  be 
adjusted  so  that  each  will  reverse  with  but  little  change  in  the 
position  of  the  bubble. 

The  adjustments  of  the  focus  of  the  telescope,  of  the  colli- 
mation,  and  for  bringing  the  middle  line  of  the  reticle  into  a 
vertical  plane,  should  be  made  precisely  as  for  the  astronomi- 
cal transit  (see  §§85,  86). 

The  reading  microscopes  must  be  kept  in  adjustment. 
Ordinarily  the  only  adjustment  that  will  be  found  necessary 
is  to  fit  the  microscope  to  the  eye  by  drawing  out  or  pushing 
in  the  eyepiece  until  the  most  distinct  vision  of  the  microm- 
eter lines  and  of  the  graduation  is  obtained.  Sometimes  it 
may  be  found  that  the  micrometer  lines  are  apparently  not 
parallel  to  the  graduation  upon  which  the  pointing  is  to  be 
made.  This  may  be  remedied  by  rotating  the  micrometer 
box  about  the  axis  of  figure  of  the  microscope.  If  to  do  this 


§  1 8 1.  DIRECTIONS  FOR   OBSERVING.  2OI 

it  is  necessary  to  loosen  the  microscope  in  its  supporting 
clamp,  great  caution  is  necessary  to  insure  that  the  distance 
of  the  objective  from  the  circle  graduation  is  not  changed. 

If  one  turn  of  the  micrometer  is  found  to  differ  very  much 
from  its  nominal  value,  in  terms  of  the  circle  graduation  (one 
turn  =  one  minute  for  the  instrument  described  in  §  178), 
it  may  be  restored  to  its  nominal  value  by  changing  the  dis- 
tance of  the  objective  from  the  circle  graduation.  An 
inspection  of  Fig.  21,  and  the  corresponding  text,  §  135,  will 
show  that  for  a  given  microscope  the  nearer  the  objective  is  to 
the  graduation  the  smaller  is  the  value  of  one  turn,  and  vice 
versa.  A  change  in  this  distance  also  necessitates  a  change  in 
the  distance  from  the  objective  to  the  micrometer  lines — these 
lines  and  the  graduation  being  necessarily  at  conjugate  foci  of 
the  objective.  This  adjustment  of  the  micrometer  value  is  a 
difficult  one  to  make,  and  so  should  not  be  attempted  unless 
it  is  certainly  necessary.  Once  well  made,  it  usually  re- 
mains sufficiently  good  for  a  long  period. 

Directions  for  Observing. 

181.  The  azimuth  mark  and  the  instrument  fix  the  two 
ends  of  a  line  of  which  the  azimuth  is  to  be  determined. 
The  azimuth  mark  must  be  placed  so  far  away  from  the 
instrument  that  no  change  from  the  sidereal  focus  of  the  tele- 
scope will  be  required  to  give  a  well-defined  image  of  it.  One 
mile  will  usually  be  found  sufficient.  A  convenient  mark  is 
a  bull's-eye  lantern  shining  through  a  small  hole  in  a  box 
which  serves  to  protect  it  from  the  wind.  An  ordinary 
tubular  lantern  mounted  in  that  way  has  been  found  satisfac- 
tory on  distances  from  one  to  three  miles.  The  size  of  the 
hole  must  be  suited  to  the  distance  and  telescope.  Too  large 
a  hole  gives  a  blur  of  light  instead  of  a  well-defined  point. 
Too  small  a  hole  makes  the  light  appear  too  faint.  A 


202  GEODETIC  ASTRONOMY.  §  1 82. 

diameter  of  one  inch  has  been  preferred  by  the  writer  at  dis- 
tances of  one  to  three  miles  for  observations  with  a  telescope 
having  an  objective  45  mm.  in  diameter  and  an  eyepiece 
magnifying  30  diameters.  Most  observers  prefer  a  much 
smaller  hole.  If  the  distance  to  the  mark  exceeds  five  miles, 
a  stronger  light,  with  a  parabolic  reflector  behind  it  or  a  lens 
in  front  of  it,  may  be  required.  Black  and  white  stripes  on 
the  front  of  the  box,  or  a  pole  accurately  in  line,  may  serve 
as  a  target  for  the  measurements  in  daylight  of  the  horizontal 
angles  necessary  to  connect  with  a  triangulation.  A  ground 
mark  (stake,  bolt,  or  stone  monument)  must  always  be  pro- 
vided to  hold  the  point  in  case  the  box  and  light  are  acci- 
dentally disturbed.  If  the  line  of  sight  passes  near  the 
ground,  the  light  usually  appears  more  unsteady  than  if  the 
line  is  high  above  the  ground.  A  long  cut  through  woods 
along  the  line  of  sight,  or  the  presence  of  objects  very  near 
to  the  line  on  either  side,  tends  to  make  the  light  appear 
unsteady,  and  introduces  a  liability  to  a  small  constant  error 
in  the  observed  azimuth.  The  direction  to  the  azimuth  mark 
is  immaterial,  except  when  a  micrometer  is  to  be  used  as 
indicated  in  §  205. 

182.  Occasionally  it  is  difficult  to  find  a  satisfactory  loca- 
tion for  an  azimuth  mark.  For  example,  the  astronomical 
station  may  be  one  of  the  stations  of  a  triangulation  located 
on  the  flat  top  of  a  mountain  in  such  a  position  that  none  but 
very  near  or  very  distant  points  are  visible  from  it.  In  such 
a  case  one  may  resort  to  a  collimator  for  an  azimuth  mark. 
The  collimator  is  an  auxiliary  telescope  rigidly  mounted  so 
as  to  face  the  instrument,  and  adjusted  to  sidereal  focus. 
The  instrument  telescope  being  also  at  sidereal  focus  and 
pointed  upon  the  collimator,  the  two  object-glasses  then  being 
toward  each  other,  the  lines  of  the  reticle  of  the  collimator 
may  be  seen  as  if  they  were  at  an  infinite  distance,  for  rays  of 


§  1 85.  DIRECTIONS  FOR   OBSERVING.  2O3 

light  proceeding  from  them  are  parallel  rays  in  the  space 
between  the  two  object-glasses.  The  middle  line  of  the 
reticle  then  defines  a  fixed  direction  from  the  instrument  and 
serves  the  same  purpose  as  the  ordinary  distant  azimuth 
mark,  provided  the  collimator  remains  fixed  in  direction. 
Great  care  is  necessary  in  mounting  a  collimator  and  in  pro- 
tecting it  to  insure  that  this  last  condition  is  sufficiently  well 
satisfied. 

183.  In  accurate  azimuth  work,  to  insure  that  errors  of 
time  and  latitude  have  but  little  effect  upon  the  computed 
azimuths,  only  close  circumpolar  stars  should  be  observed. 
It  will  be  found  advisable  to  use  only  the  four  close  circum- 
polars    of    which    the    apparent    places    are    given     in     the 
Ephemeris,  namely,  a,  A,  and  #  Ursae  Minoris  and  5  I  Cephei. 
Fig.  23,  showing  their  relative  positions,  will  be  found  a  con- 
venient aid  in  finding  and  identifying  them.     The  figure  also 
shows  roughly  their  right  ascensions,  declinations,  and  mag- 
nitudes.    The  position  of  the  figure  on  the  page  corresponds 
to  the  position  of  the  stars  in  the  sky  at  oh  sidereal  time. 
The  figure  in  the  sky  rotates  once  around  in  a  counter-clock- 
wise direction  every  twenty-four  sidereal  hours.      The  arrow 
at  the  pole  indicates  by  its  length  and  direction  the  apparent 
motion  of  the  pole  among  the  stars  in  a  century. 

184.  Three  methods  of  observing  will  be  treated.     First, 
that  in  which  a  close  circumpolar  star   is  observed  at  any 
hour-angle,  and  the  instrument   is  used  as  a  direction  instru- 
ment; second,  a  similar  method  in  which  the  instrument  is 
used  as  a  repeater;   and  third,  a  method  of  observing  upon  a 
close  circumpolar  near  elongation  with  an  eyepiece  micrometer 
only,  without  using  the  horizontal  circle. 

185.  For   the    first    method    the    following   program    of 
observing  may  be  used :  Point  upon  the  mark  and  read  the 
horizontal  circle,  twice  each;  point  approximately  upon  the 
star  and   place   the  striding  level   in   position;    perfect    the 


204  GEODETIC  ASTRONOMY.  §  185. 

pointing  upon  the  star  and  note*  the  chronometer  time  of 
the  bisection;  read  the  horizontal  circle;  point  again  upon 
the  star,  noting  the  chronometer  time;  read  the  circle;  read 
the  striding  level  and  reverse  it;  point  upon  the  star  and  note 
the  chronometer  time;  read  the  circle;  read  and  remove  the 
striding  level;  point  upon  the  mark  and  read  the  circle,  twice 
each.  This  completes  a  half-set.  Reverse  the  telescope  in 
altitude,  and  the  instrument  in  azimuth,  and  repeat  the  same 
routine  for  a  similar  half-set.  The  whole  set  will  thus  be 
made  up  of  six  pointings  upon  the  star  and  eight  pointings 
upon  the  mark,  with  the  corresponding  circle  readings,  and 
four  readings  of  each  end  of  the  bubble  of  the  striding  level. 
It  will  be  found  a  convenience  when  making  the  computation 
if  the  altitude  of  the  star  is  read  to  the  nearest  minute  from 
the  vertical  circle  once  for  each  half-set.  The  special  consid- 
erations that  lead  to  the  recommendation  of  the  above  routine 
are:  that  the  effect  of  twisting  of  the  instrument  in  azimuth 
should  be  eliminated  from  the  result  for  each  half-set  as  far 
as  possible;  that  the  level  bubble  should  have  time  to  settle 
without  delaying  the  observer  for  that  purpose;  and  that  the 
observations  of  a  set  should  be  completed  as  quickly  as  possi- 
ble to  avoid  the  effects  of  instability  of  instrument.  More 
pointings  in  a  set  would  serve  to  decrease  the  effect  of  errors 
of  observation,  but  tend  to  increase  the  errors  due  to  in- 
stability. 

To  secure  accurate  results,  the  pointings  upon  the  star, 
mark,  and  graduations  must  be  carefully  made;  all  heating 
of  the  instrument  above  the  temperature  of  the  outside  air, 
and  especially  all  unequal  heating  of  its  parts,  must  be  avoided 
as  far  as  possible;  the  manipulation  must  be  made  with  as 
little  applied  force  as  possible,  especially  at  the.  instants  when 

*  The  allowable  error  in  time  being  comparatively  large  in  azimuth 
work,  the  observer  may  simply  call  "Tip"  at  bisection  and  let  an  assistant 
read  the  chronometer. 


EXAMPLE  OF  RECORD. 


205 


bisections  are  being  made.  Other  conditions  being  unchanged, 
the  more  rapidly  the  observations  are  made  the  greater  the 
accuracy,  because  the  errors  due  to  instability  are  smaller. 

186.  Before  commencing  the  next  set  of  observations,  the 
graduated  circle  should  be  shifted  to  another  position  so  that 
each  microscope  will  come  over  a  different  part  of  the  gradua- 
tion.    To  insure  as  complete  an  elimination  of  periodic  errors 
of  graduation  as  possible,  it  is  advisable  to  shift  the  circle 
between  the  successive  sets  of  a  series  so  that  the  various 
positions  of  a  given  microscope,  when  pointing  upon  the  mark 
in  a  given  position  of  the  telescope,  shall  divide  the  interval 
between  successive  microscopes  (120°  or  180°,  usually)  into 
as  many  equal  parts  as  there  are  sets  in  the  series. 

187.  EXAMPLE    OF    RECORD. 

Station — West  Base.  Observer — A.  F.  Y. 

4>  =  41°  29'  03". 5  Instrument — T.  &  S.  Altazimuth  No.  72. 

Date— Sept.  13    1877.  i  div.  of  striding  level  =  2". 12. 

Star — d  Ursae  Minoris.  Chronometer — Negus  1431  (Sidereal). 

a  =  i8h  nm  47*.  5  Chronometer  correction  =  +  4*.  5. 

d  =  86°  36'  4i".o 


H 
"o 

Level 
Readings. 

Horizontal  Circle. 

Object. 

c 

Chronometer 
Times. 

1 

W. 

E. 

Index. 

Mic.  A. 

Mic.  B. 

Mic.  C. 

IX 

t      d 

d 

d 

d 

d 

d 

Mark 

D 

142°  25' 

3     19-7 

19.0 

20.5 

20.  o 

21.7 

20.5 

*4 

44 

142    25 

3*  21. 

19.7 

20.0 

21.0 

20.0 

18.3 

Star 

44 

d 

d 

Oh  ogm  OQ..O 

158       20 

o    24. 

23.0 

25-0 

26.5 

24-5 

24-3 

44 

60.3 

47-1 

09    oi  .0 

158       20 

o    26. 

25-5 

27-5 

28.2 

117.1 

26.7 

44 

65.6 

09  50  .5 

158       20 

o    26. 

26.8 

28.0 

28.8 

28.0 

28.2 

Mark 

44 

142       25 

3     23. 

22.0 

22.0 

23.0 

22.4 

2O.  2 

*4 

142       25 

3      22. 

21.4 

20.7 

22.  O 

21.4 

18.9 

R 

322       25 

3     °9- 

08.  I 

07-3 

08.8 

08.4 

06.2 

kfc 

322       25 

3     07- 

06.6 

°5-3 

07.2 

06.3 

04.7 

Star 

*; 

0      20      03   .5 

338       20 

o    56. 

55-2 

57-5 

58.2 

55-7 

M 

ti 

62.4 
46.0 

21     33  -5 

22      46   .0 

338      20 
338      20 

i     07. 

I    19. 

06.6 
18.9 

08.3 
20.7 

09.2 
2O.9 

07-H 
19.0 

07.4 
19.0 

Mark 

44 

322      25 

3    07.8 

07.4 

06.4 

07.9 

07.6 

05.5 

322      25 

3     07.9 

07.4 

06.2 

08.2 

07.6 

06.5 

The  mean  zenith  distance  of  the  star  during  the  observations,  from  two  approximate  read- 
ings of  the  vertical  circle,  was  found  to  be  40°  52'. 

*  The  reading  of  the  whole  turns,  for  the  other  micrometers,  is  not  repeated  in  the  record, 
but  in  making  the  readings  it  is  called  out  to  the  recorder.  If  he  finds  it  the  same  as  for 
micrometer  A,  the  record  is  as  above.  If  it  falls  a  unit  below  that  for  A,  he  indicates  it  by 
writing  a  minus  sign  over  the  recorded  reading  of  the  head;  and  if  it  is  a  unit  above,  he  calls 
attention  to  it  and  records  it  as  60  +  the  given  reading. 


206  GEODETIC  ASTRONOMY.  §  1 88. 


The  Circle  Reading. 

188.  In  the  record  above,  two  readings  are  given  for  each 
position  of  each  reading  microscope.  When  commencing  to 
read  microscope  A  for  the  first  time,  for  example,  in  the  above 
set  of  observations,  the  field  of  view  looked  as  shown  in 
Fig.  25.*  The  reading  was  evidently  25'  plus  the  angle 
represented  by  the  interval  from  the  zero  of  the  microscope 
(the  position  in  which  the  micrometer  lines  are  shown)  to  the 
25'  graduation.  This  plus  quantity  is  read  directly  from  the 
micrometer  when  the  25'  graduation  is  bisected  (called  the 
forward  reading),  namely,  3'  1 9". 7,  provided  there  is  no  error 
of  pointing,  and  provided  each  turn  of  the  micrometer  repre- 
sents exactly  \' .  But  neither  of  these  conditions  are  realized 
in  practice,  and  a  more  reliable  result  may  be  secured  if  a 
reading  is  also  made  on  the  30'  graduation  (called  the  back- 
ward reading).  With  perfect  pointing  and  perfect  adjustment 
of  the  micrometer,  the  screw  would  necessarily  be  turned 
exactly  five  revolutions  backward  to  pass  from  a  pointing  on 
the  25  line  to  a  pointing  on  the  30  line,  and  the  reading  of 
the  head  of  the  micrometer  would  be  the  same  in  both  cases. 
It  is  actually  0.7  less.  Neglect  for  a  moment  all  considera- 
tion of  possible  errors  in  pointing  and  reading.  These  two 
readings  would  then  indicate  that  one  turn  of  the  micrometer 

c' 
represents!  =  59". 856.      Hence  the  measured  interval 

of  3'  19". 7  from  the  zero  of  the  micrometer  to  the  25'  line 
represents  ^3-|^j(59//.856)  =  3'  19". 2.    This  procedure  does 

*  The  field  of  view  is  here  shown  as  it  actually  appears  to  the  observer. 
The  microscopes  invert,  and  therefore  the  graduation  really  increases  in 
the  opposite  direction  from  that  here  shown. 

f  0.7  division  =  0.012  turn.     (6od  =  i  turn.) 


§  189.  THE    CIRCLE  READING.  2O? 

not  involve  any  assumption  as  to  the  exact  value  of  one  turn, 
but  in  fact  derives  the  circle  reading  from  each  pair  of  mi- 
crometer readings  upon  the  assumption  merely  that  the  grad- 
uated interval  on  the  circle  is  exact.  This  process  of  making 
the  correction  for  the  run  *  of  the  micrometer  may  be  put  in 
convenient  form  for  rapid  computation  for  the  above-described 
instrument  as  follows.  The  above  figure  and  explanation  will 
serve  as  a  sufficient  proof  of  the  formulae  given. 

189.  Let  F'  be  the  forward  reading  of  the  micrometer, 
both  comb  and  head,  expressed  in  turns,  this  reading  being 
taken  on  the  line  of  the  graduation  which  is  adjacent  to  the 
zero  of  the  micrometer  in  the  direction  of  increasing  readings 
of  the  micrometer;  and  let  F  be  the  corresponding  reading  of 
the  micrometer  head,  expressed  in  turns.  Let  B  be  the 
backward  reading  of  the  micrometer  head,  expressed  in  turns, 
taken  on  that  line  of  the  graduation  which  is  adjacent  to  the 
zero  of  the  micrometer  in  the  opposite  direction.  Let  the 
true  reading  of  the  circle  to  be  derived  be  called  T.  Let 
the  interval  between  lines  of  the  graduation  be  called  /  (5' 
in  the  preceding  illustration).  Then  one  turn  of  micrometer 
'/  5' 


F-H-S+F-J3- 


Strictly,  the  required  value  of 


r' 
T  is    then    T  =  —  \     &  _  gC^O*      But    remembering   that 

F—  B  is  ordinarily  only  one  or  two  sixtieths  of  a  turn,  and 
is  therefore  small  as  compared  with  the  complete  interval  (5 

turns),  we  may  write  T=  —         —(F')  =  (i'  +  —  ^)  (F'), 
in  which  the  difference  B  —  F  is  now  taken  in  divisions  of 


*  The  run  of  a  micrometer  is  the  amount  by  which  one  turn  exceeds,  or 
falls  short  of,  its  nominal  value, — 0.7  in  the  above  example.  The  error  of 
runs  is  the  error  in  the  result  which  is  introduced  by  neglecting  the  run  of 
the  micrometer. 


2O8  GEODETIC  ASTRONOMY.  §  1  90. 

the  head,  and  considered  to  represent  seconds.  The  nominal 
reading  of  the  micrometer  being  (i7)  (F'),  the  correction,  Crt 
to  this  nominal  reading,  or  correction  for  run  to  be  applied 
to  the  forward  reading,  is 


(88) 


The  values  of  Cr  will  be  found  tabulated  in  §  309  for  the 
arguments  F',  the  nominal  forward  reading,  and  B  —  F,  ex- 
pressed in  divisions,  or  nominally  in  seconds.  This  table 
applies  of  course  only  to  a  reading  microscope  of  the  above 
type  in  which  five  turns  are  nominally  equal  to  5',  one  space 
of  the  circle  graduation,  and  each  division  of  the  head  is 
nominally  i".  The  table  may  be  used  to  correct  each  forward 
reading  of  a  series,  or  the  mean  value  of  Cr  taken  out  from 
the  table  for  each  reading  may  be  applied  to  the  mean  of  the 
forward  readings.  A  similar  table  may  be  constructed  on  the 
same  principle  for  any  other  micrometer. 

190.  In  developing  the  preceding  method  of  computing 
the  true  reading  of  the  circle,  the  accidental  errors  of  pointing 
upon  the  graduation  have  been  entirely  ignored,  it  being 
tacitly  assumed  that  they  are  small  as  compared  with  the 
error  of  runs.  Let  us  now  make  the  converse  supposition  — 
that  the  error  of  runs  is  small  as  compared  with  the  error  of 
pointing.  On  this  supposition  the  forward  and  back  readings 
of  the  head  differ  simply  because  of  errors  of  pointing. 
Hence  they  are  equally  good  determinations  of  the  seconds 
of  the  reading,  and  their  mean  is  to  be  taken.  On  this  sup- 
position the  true  reading,  so  far  as  the  seconds  are  concerned, 
is 

.    ,    .     .     .     ..'.     (89) 


§  IQ2.  LEVEL    CORRECTION.  209 

191.  The  use  of  (89)  instead  of  (88),  or  the  corresponding 
tables,  leads  to  quite  a  saving  of  time.     Two  other  considera- 
tions also  point  to  such  use  as  advisable.     Firstly,  the  true 
result  sought  is  in  reality  between  the  results  given  by  these 
two  methods,  since  errors  of  run  and  errors  of  pointing  both 
exist,  and  in  general  neither  are  insensible  as  compared  with 

F' 

the  other.  •  Secondly,  —  in  (88)  is  as  apt  to  be  greater  than 

Fr  . 
J  as  it  is  to  be  less  than  £.     If  —  is  J,  the  use  of  (88)  gives 

numerically  the  same  result  as  (89).  Hence  the  results  from 
the  use  of  (88)  are  as  apt  to  be  greater  as  to  be  less  than  those 
from  (89),  and  the  greater  the  number  of  observations  treated 
the  nearer  the  results  from  the  two  formulae  agree.  Hence, 
in  general,  there  is  not  a  sufficient  gain  in  accuracy  over  the 
procedure  indicated  in  (89)  to  justify  the  time  required  to 
correct  for  errors  of  run.* 

The  Level  Correction. 

192.  Any  inclination  of  the  horizontal  axis  affects  the 
circle  reading  corresponding  to  the  pointing  upon  the  star, 
and  necessitates  a  correction  which  is  to  be  determined  from 
the  readings  of  the  striding  level. 

In  Fig.  26  let  NES  W  represent  the  horizon.      Let  s  be 
the  star,  and  Z  the  zenith.      If  the  instrument  is  in  perfect 

*'  Sometimes  the  mean  value  for  the  run  of  a  given  micrometet  is  de- 
rived from  a  special  series  of  observations  for  that  purpose:  the  run  is 
assumed  to  be  constant;  and  a  correction  based  upon  this  mean  value  is 
applied  to  the  mean  results  computed  by  (89).  This  procedure  shortens 
the  work  of  applying  the  correction  for  run,  after  the  mean  value  of  the 
run  has  been  computed.  The  validity  of  the  assumption  that  the  run  is  a 
constant  is  so  doubtful,  however,  that  it  seems  that  if  the  correccion  foi 
run  is  to  be  applied  at  all,  it  should  be  based  upon  a  value  for  the  run  de- 
rived from  the  very  readings  that  are  to  be  corrected. 


2IO  GEODETIC  ASTRONOMY.  §  IQ2. 

adjustment,  when  the  telescope  is  pointed  upon  the  star  the 
plane  in  which  the  telescope  is  free  to  swing  about  its  hori- 
zontal axis  is  defined  by  the  arc  ZsP,  in  which  P  is  the  pole 
of  the  great  circle  passing  through  Z,  and  A,  the  point  in 
which  the  horizontal  axis  produced  pierces  the  celestial 
sphere.  If,  now,  the  horizontal  axis  be  given  an  inclination 
b,  the  west  end  being  placed  too  high,  A  will  move  to  a  point 
A' ,  at  a  distance  b  along  Az.  The  zenith  of  the  instrument 
will  virtually  be  shifted  to  Z'  (such  that  the  arc  AZZ'  =  90°, 
and  ZZ'  =•  &),  that  being  the  nearest  point  to  the  true  zenith 
to  which  the  telescope  can  be  pointed.  The  telescope  is  now 
free  to  swing  in  the  arc  Z'P.  But  this  arc  does  not  pass 
through  s.  To  bisect  the  star  it  is  necessary  to  turn  the 
instrument  about  its  vertical  axis,  which  now  passes  through 
Z\  until  the  telescope  swings  in  the  arc  Z's.  A'  will  then 
be  in  such  a  position  as  A" ' .  The  change  in  the  circle  read- 
ing, due  to  the  inclination  of  the  axis,  is  evidently  measured 
by  the  angle  PZ's.  The  circle,  if  graduated  clockwise,  now 
reads  too  small  by  that  amount,  which  will  be  called  CL. 
Consider  the  spherical  triangle  Z'Ps.  In  this  triangle  the 
angle  at  Z'  is  the  required  CLJ  the  angle  at  P  is  b,  the  side 
Ps  is  the  altitude  of  the  star  A,  and  the  side  Z's  is  the  zenith 
distance  of  the  star  as  measured  with  the  displaced  instru- 
ment. But,  the  displacement  being  small,  Z's  may  be  for 
the  present  purpose  considered  equal  to  Zs,  the  zenith  dis- 
tance, C,  of  the  star.  From  the  proportionality  of  the  sines 
of  angles  and  opposite  sides  in  the  triangle  Z'Ps,  we  may 

sin  CL       sin  b  .  . 

write  — — -r-  =  — — -.      Replacing  sin  CL  and  sin  b  by  CL  and 
sin  A        sin  C 

b,    those   angles  being  small,   and    solving   for   CL,   there   is 
obtained 


§  193-  FORMULA.  211 

Expressing  the  inclination  b  in  terms  of  the  readings  of 
the  striding  level,  there  is  obtained  the  complete  formula  for 
the  level  correction, 

CL  =  \(w  +  «0  -  (e  +  S)\j  tan  A,      ,     ',     (91) 

for  a  level  having  its  divisions  numbered  both  ways  from  the 
middle. 

CL=\(w  +  e)-  (W  +  O}  j  tan  A  .     .     .     (92) 

4 

for  a  level  numbered  continuously  in  one  direction,  the 
primed  letters  referring  to  the  readings  taken  in  the  position 
in  which  the  numbering  increases  toward  the  east.  C L  as 
given  by  these  formulae  is  the  correction  to  the  circle  reading 
on  the  supposition  that  the  numbers  on  the  circle  graduation 
increase  in  a  clockwise  direction. 

Similar  corrections  to  the  circle  readings  upon  the  mark, 
derived  from  corresponding  readings  of  the  striding  level,  are 
necessary  if  the  line  of  sight  to  the  mark  is  much  inclined. 
Ordinarily  the  line  of  sight  to  the  mark  is  so  nearly  horizontal 
that  such  corrections  are  negligible,  and  the  corresponding 
level  readings  may  be  dispensed  with,  provided  that  care  is 
taken  to  keep  the  instrument  well  levelled  up. 

Azimuth  of  the  Star. 

193.  The  preceding  formulae  suffice  for  the  computation 
of  the  horizontal  angle  between  the  star  and  mark.  It 
remains  to  compute  the  azimuth  of  the  star. 

The  detail  of  the  process  of  computing  the  hour-angle  of 
the  star  from  the  chronometer  reading  need  not  be  stated 
here.  The  hour-angle  t  and  the  declination  8  of  the  star 
being  known,  as  well  as  the  latitude  of  the  station  0,  the 
azimuth  of  the  star  may  be  computed  from  the  spherical 


212  GEODETIC  ASTRONOMY.  §  193. 

triangle  defined  by  the  star,  the  zenith,  and  the  pole.  Certain 
sides  and  angles  of  this  spherical  triangle  have  the  values 
indicated  in  Fig.  9,  in  terms  of  the  angles,  z  the  azimuth  of 
the  star  reckoned  from  the  north,  A  its  altitude,  and  /,  0, 
and  d.  From  the  principle  that  the  sines  of  the  angles  of  a 
spherical  triangle  are  proportional  to  the  sines  of  the  opposite 
sides,  we  may  write 

sin  z        sin  / 


Also,  from  the  principle  that  in  any  spherical  triangle  the 
cosine  of  any  side  is  equal  to  the  product  of  the  cosines  of  the 
other  two  sides  plus  the  product  of  their  sines  into  the  cosine 
of  the  opposite  angle,  we  may  write  the  two  formulae, 

sin  tf  =  sin  0  sin  A  -f-  cos  0  cos  A  cos  z  ;       .     (94) 
sin  A  =  sin  0  sin  d  +  cos  0  cos  d  cos  t.    .     .     (95) 

By  substituting  sin  A  from  (95)  in  the  first  term  of  the  second 
member  of  (94),  and  solving  the  resulting  equation  for  cos  Z, 
there  is  obtained 


(i  —  sin*  0)  sin  £  —  sin  0  cos  0  cos  d  cos  t 


cos 


(96) 


By  substituting  cos9   0  for   i  —  sin2  0,    and   dividing  both 
numerator  and  denominator  by  cos  0,  (96)  reduces  to 


cos  0  sin  #  —  sin  0  cos  tf  cos  t  ,     x 

-  cos  .4 '   '     '     (97) 


§195-  CURVATURE   CORRECTION.  21$ 

If  (93)  is  now  divided  by  (97),  and  both  denominators  of  the 
resulting  equation  are  divided  by  cos  6,  there  results 

sin  t 

tan  z  =  -7— ^ : — -7 -,        .     .     (98) 

cos  0  tan  o  —  sin  0  cos  / 

from  which  z  may  be  computed  from  the  known  values  of  0, 
<?,  and  t.  In  (98)  z  is  the  azimuth  of  the  star  east  or  west, 
respectively,  of  north,  as  the  hour-angle,  is  reckoned  from 
upper  culmination  to  the  eastward  or  westward,  respectively. 

The  Curvature  Correction. 

194.  To  apply  this  formula  to  the  star  for  each  pointing 
made  upon  the  star  during  a  set  of  observations  would  be  too 
laborious  a  process.      If  for  the  t  of  the  formula  is  taken  the 
mean  of  the  hour-angles  of  the  set,  the  computed  azimuth  is 
that  corresponding  to  the  mean  hour-angle,   but  is  not   the 
required  mean  of  the  azimuths  corresponding  to  the  separate 
hour-angles,  since  the  rate  of  change  of  the  azimuth  is  con- 
tinually varying.     The  difference  between  the  two  quantities 
indicated  by  the  italics  is  small,  though  not  usually  negligible, 
for  the  interval  of  time  covered  by  a  set  of  observations.     It 
is  proposed  to  derive  a  sufficiently  precise  expression  for  this 
difference,    and    to  apply   it   as  a    correction    to    the    result 
obtained  by  using  a  mean  value  of  /  in  formula  (98). 

195.  Let  /^   /„   /„  .  .  .  /„  be  the    observed  hour-angles, 
and    z»    z»    **  .  .  .  zn    the    respective    corresponding    azi- 
muths.    Let   t0  be   the  mean  of  the  observed  hour-angles, 

= '  '  '  ' — -,  and  let  z0  be  the  azimuth  correspond- 
ing to  /..  Let  At,  =  t,  -  /.,  At,  =  /,-*„,...  Atn  =  /„  -  /o. 
Then 

At,  +  At,  +  J/a  .  .  .  AtH  =  o.       ...     (99) 


2I4 


GEODETIC  ASTRONOMY. 


§196. 


If  the  third  and  following  derivatives  of  z  with  respect  to 
t  be  neglected,  we  may  write 


dz 

dz 
~dt 

dz 


d*z 


-  .     .     .     (100) 


in  which  -=-  and  -TT  are  the  first  and  second  derivatives  corre- 
at          at 

spending  to  z0  and  /„.     The  mean  value  of  z  is 


dz 

The  terms  of  the  form  ~r.At  disappear  by  virtue  of  the  rela- 
tion expressed  by  (99). 

196.  An  approximate  value  of  -r^-  may  be  derived  from 

equation  (98)  as  follows:  Since  only  close  circumpolar  stars 
are  being  considered,  z  is  always  small,  and  as  an  approxi- 
mation we  may  write  z  (in  arc  measure)  for  tan^.  Also, 
since  for  the  stars  under  consideration  8  is  nearly  90°,  cos  <j> 
tan  d  is  much  larger  than  sin  0  cos  /,  and  the  quantity 

— : — -T may  be  assumed  constant  for  the 

cos  0  tan  6  —  sin  <p  cos  / 

interval  of  a  few  minutes  over  which  the  observations  of  a  set 
extend,  and  will  be  called  C.  Equation  (98)  becomes  after 

i  dz 

these    substitutions  z  =  C  sin  /,   whence  -j-  =  C  cos  /,  and 

at 


§  197-  CURVATURE   CORRECTION.  21$ 

d*z  sin  t 

-r*    —  —  C  sin  t  =  --  —  -  -  -  --  :  -  =  —  tan  2. 

at  cos  0  tan  d  —  sin  0  cos  / 

d*z 
For  substitution  in  (101)  this  must  be  written  -=-^  =  —  tan  ZQ, 

according  to  the  statement  following  equations  (100).      Equa- 
tion (101)  then  becomes 


fa'-  (I02) 


197.   In  deriving  the  expression  —  tan  z9  for   -JT,   z  was 

considered  to  be  in  arc  measure.  Therefore,  putting  At^ 
At^  At^,  .  .  .  Atn  also  in  arc  in  that  term,  and  then  dividing 
the  whole  term  by  sin  i"  to  reduce  to  seconds,  it  becomes 


sm 

Finally,  to  facilitate  the  computation  by  enabling  the 
computer  to  use  the  table  given  in  §  307,  we  may  write 
At  (in  arc)  —  2  sin  J///,  or  %At  —  2  sin'  \At. 

Equation  (102)  may  then  be  written 

Zi-f-22-f-zs. .  .zn  i  /2  sin2  \Ati       2  sin2  \Ati  2  sin*  A^/»A 

-IT         =Zo-t&nZ°n(--^^+-^i—  +  ''--^^)'  <I03) 


in  which  the  values  of  the  terms : — =77—,  etc.,  are  known 

from  the  table  given  in  §  307.  Certain  approximations  have 
been  made  in  deriving  the  last  term  in  (103),*  but  the  total 
value  of  that  term  is  so  small  that  the  errors  due  to  these 
approximations  are  negligible. 

*  For  a  slightly  different  method  of  deriving  the  same  formula,  see 
Doolittle's  Practical  Astronomy,  pp.  537,  538. 


2l6  GEODETIC  ASTRONOMY.  §  199. 


Correction  for  Diurnal  Aberration. 

198.  To  the  results  as  computed  by  the  above  formulae 
there  must  still  be  applied  a  small  correction  for  the  effect  of 
diurnal    aberration.     Because    of    the    rapid    motion    of    the 
observer  due  to  the  rotation  of  the  Earth  on  its  axis,  the  star 
is    seen   slightly   displaced   from    its    real   position,   and    the 
apparent  azimuth  of  the  star  is  correspondingly  affected. 

Suppose  that  at  the  instant  of  observation  the  station  from 
which  the  observation  is  made  is  moving  in  the  direction  AB i 
of  Fig.  27.  AB  necessarily  passes  through  the  east  point,  on 
the  celestial  sphere,  of  the  observer's  horizon  at  that  instant. 
Let  SA  be  the  true  direction  of  the  ray  of  light  from  the  star. 
The  figure  is  drawn,  then,  in  the  plane  defined  by  the  star, 
the  observer,  and  the  east  point  of  the  observer's  horizon. 
In  consequence  of  the  aberration  the  star  will  be  seen  in  the 
direction  AS' .  Let  CA  and  AD  be  drawn  proportional  to  the 
distance  V  traversed  by  a  ray  of  light  in  one  second,  and  the 
distance  v  traversed  by  the  station  of  observation  in  one 
second,  respectively.  Complete  the  parallelogram  CADG. 
Call  the  angle  SAB  ft,  and  the  angle  CAG  =  SAS'  dft. 
dp  is  the  apparent  displacement  of  the  star  measured  in  the 

plane  of  the  figure.    From  the  triangle  CAG,  —. — —  =  j^,  or 

dft  =  sirr'-p:  sin  ft. 

Substituting  for  v,  0.288  cos  0,  and  for  V  186000  (see 
§  96),  there  is  obtained 

dft  =  o".3i9  cos  0  sin  ft Oo4) 

199.  It  remains  to  determine  the  effect  of  this  displace- 
ment upon  the  star's  azimuth.     In  Fig.  28   let  NESW  be 


§  199.        CORRECTION  FOR  DIURNAL  ABERRATION.  21J 


the  horizon  of  the  observer,  Z  his  zenith,  and  s  the  star. 
Prolong  the  great  arc  Zs  to  F.  In  the  spherical  triangle  sFE 
the  angle  at  F  is  90°  ;  the  side  sF  is  the  altitude  of  the  star  A, 
the  side  FE  is  90°  —  z  (z  is  the  azimuth  of  the  star),  and  the 
side  sE  is  the  /3  of  the  preceding  paragraph.  Call  the  angle 
sEF  v.  Then  from  Napier's  rules  fora  right  spherical  tri- 
angle we  may  write 

cos  A  sin  FsE  =  cos  v  ;    ....     (105) 

sin  FsE  sin  ft  =  cos  z  ;     ....     (106) 

cos  A  sin  z  =  cos  ft  .....     (IO7) 

Substituting  the  value  of  sin  FsE  from  (106)  in  (105),  there  is 
obtained 

cos  A  cos  z  =  sin  ft  cos  v.  (IQ8) 

We  require  the  effect  upon  z  of  a  small  change  in  ft. 
This  may  be  found  by  differentiating  (107)  and  (108)  with 
respect  to  A,  z,  and  ft  (but  not  with  respect  to  v,  since  that 
is  not  changed  by  the  change  in  ft).  Thus  we  find 

cos  A  cos  zdz  —  sin  A  sin  zdA  =  —  sin  ftdfi  ;       (109) 
—  cos  A  sin  zdz  —  sin  A  cos  ^^4  =  cos  ft  cos  «/^S.  (no) 

Multiply  (109)  by  cos  z  and  (no)  by  sin  z,  and  subtract  the 
second  of  the  resulting  equations  from  the  first,  and  there  is 
obtained 

cos  Adz  =  —  cos  z  sin  ftdft  —  sin  z  cos  ft  cos  vdft.  (in) 

Solve  for  dz,  multiply  both  numerator  and  denominator  of 
the  resulting  fraction  by  sin  ft,  and  eliminate  cos  v  by  substi- 
tuting its  value  from  (108),  and  the  result  is 


.  . 

sin  ft  cos  A 


dft.  (112) 


218  GEODETIC  ASTRONOMY.  §  2OO. 

Substituting  the  value  of  dft  from  (104),  we  have,  finally, 

COS  0  COS  2 


„ 

=   _o'.3i9 


s 


For  a  close  circumpolar  -  -r  will  always  be  nearly  unity, 

and  so  will  be  cos  zt  except  for  stations  very  near  the  pole. 
Hence  for  such  stars  it  is  nearly  exact  to  use 

d*=  —  o".32  ......     (114) 

The  greatest  variation  from  this  value  for  the  four  circum- 
polars  mentioned  in  §  183,  and  for  stations  below  latitude 
50°,  is  o/r.O2.  Hence  for  most  purposes  it  suffices  to  use 
(i  14),  that  is,  to  dispense  with  the  computation  of  the  factor 

—  -  -  -.  —  of  (113).     That  will  be  done  in  the  examples  of 
cos  A 

this  book. 

The  sign  in  (i  13)  and  (114)  is  for  the  correction  when  the 
azimuth  of  the  star  is  expressed  as  an  angle  west  of  north. 
If  the  azimuth  of  the  star  is  expressed  as  an  angle  east  of 
north,  the  sign  must  be  changed  to  -{-. 

Example  of  Computation. 

200.  The  computation  of  the  set  of  observations  given  in 
§  187  is  as  follows.  No  correction  was  applied  for  run  of 
micrometers,  the  true  reading  of  the  circle  being  derived  by 


For  First  Half-set. 

Mean  circle  reading  on  star  ..........................  =158°  20'    26".  38 

Level  correction  =  (101.8  -  112.7X0.530X0.865)  ........  =  -  5  .00 

Corrected  mean  reading  on  star  ......................  =158  20     21.38 

Mean  reading  on  mark  ..............................  =  142  28     20  .88 

Mark  west  of  star  ..................................  =     15  52     00.50 


§200 


COMPUTATION. 


2I9 


For  Second  Half -set. 

Mean  circle  reading  on  star =  338°  21'    O7"»94 

Level  correction  =  (109.2  —  108.4X0.530X0.865) =                 -j-  o  .36 

Corrected  mean  reading =338     21     08.30 

Mean  reading  on  mark =  322     28     07  .15 


Mark  west  of  star, 


=    15     53    01  .15 


Mark  west  of  star  from  whole  set  (mean). 

Mean  of  observed  chronometer  times 
Chronometer  correction 


Mean  of  sidereal  times. 
a 


=     15     52     30  .82 
=    oh   I5m  I2-.4 


=    o     15     16.9 
=  18     ii     47.5 


Mean  hour-angle,  /0 ,  west  from  upper  culmination.. 


_  j  6h  03™  29*. 4 

~~   (  90°  52'  2l".0 

log  cos  0  =    9.8745614     log  sin  <t>  =  9.8211300  log  sin  /  =  9.9999497 

log  tan  S  =    1.2275941     log  cos  /0  =  8.1826260;*  log  12.66198  =  1.1025016 


1.1021555 

-f-  12.65189 
-f-   0.01009 

12.66198 

2  sin2    At 


•0037500W 
.01009 


log  tan  z  —  8.8974481 
z  =4°  30'  54"-47 


j    8.o< 
(  —  o. 

i  /2  sin2  \Ai±  \ 

o-   : —  +  ...)=         -6  . 

n  \     sin  i  / 


—  tan  00- 


20 


Correction  for  diurnal  aberration      =         —  o  .32 


sin  i 

" 

7m 

6 

S 

ii 

21 

•4 
•9 

101' 

75 
56 

'•97 
.22 

•51 

4 
6 

7 

51 
21 

33 

.1 
.1 
.6 

46 
79 

112 

.21 
.20 
.21 

6)471 

•32 

Azimuth  of  star,  west  of  north 
Mark  west  of  star 


=  4  30  47  .95 
=  15   52  30  .82 


Azimuth  of  mark,  west  of  north       =20  23  18  .77 


Mean  =  78".  55 
log  78.55  =  1-89515 
log  tan  z  —  8.89745 

Curvature       (0.79260 
correction   (    5". 20 


220  GEODETIC  ASTRONOMY.  §  2O2. 

Program  of  Observing  for  the  Method  of  Repetitions. 

201.  To   measure  a  horizontal  angle   by  repetitions   one 
must  use  an  instrument  having  a  clamp  and  tangent  screw  to 
control  the  motion  of  the  lower  or  graduated  circle,  in  addi- 
tion to  a  similar  clamp  and  tangent  screw  to  control  the  rela- 
tion between  the  upper  circle  carrying  the  verniers  and  the 
lower  graduated  circle.     At  the  beginning  of  the  measurement 
the  circle   is  read.      By   a  suitable  manipulation  of  the   two 
motions,  upper  and  lower,  the  angle  to  be  measured  is  multi- 
plied  mechanically   three,   five,   or  more   times.     The   circle 
being  read  again  gives  the  measured  value  of  the  multiple 
angle,    from    which    the    required   angle   is   readily   derived. 
This  process  serves  to  greatly  decrease  the  errors  arising  from 
erroneous  readings  of  the  verniers,  and  errors  of  graduation. 
But  any  lost  motion,  or  false  motion,  in  clamps  and  tangent 
screws,   affects  the  measured    angle  directly.     To  eliminate 
this  error  as  far  as  possible  one  may  measure  both  the  required 
angle  and  its  explement, *  always  revolving  the  instrument  in 
a  clockwise  direction  with  either  motion  loose,  and  making  all 
pointings  with  either  tangent  screw  so  that  the  last  motion  of 
that  screw  is  in  the  direction  in  which  the  opposing  spring  is 
being  compressed.    With  this  procedure,  unless  the  action  of 
the  clamps  and  tangent  screws  is  variable,  the  derived  values 
of  both  the  angle  and  its  explement  will  be  too  large  or  too 
small    by   the  same  amount.     The   mean  of    the   measured 
angle  and   360°  minus  the  measured  explement  will  be  the 
correct  value  of  the  angle  unaffected  by  the  constant  errors 
of  the  clamps  and  tangent  screws. 

202.  The    following    is    a   convenient    program     for    the 
measurement  of  an  azimuth  by  repetitions.     After  all  adjust- 
ments have  been  made  and  the  instrument  carefully  levelled, 

*  360°  minus  a  given  angle  is  called  the  explement  of  that  angle. 


§  202.  METHOD    OF  REPETITIONS.  221 

clamp  the  upper  circle  to  the  lower  in  any  arbitrary  position; 
point  approximately  upon  the  star;  place  the  striding  level 
in  position  and  read  it;  reverse  it,  read  it  again,  and  remove 
it;  point  accurately  upon  the  star,  noting  the  chronometer 
time,  and  using  the  lower  clamp  and  tangent  only;  read  the 
horizontal  circle;  unclamp  the  upper  motion  and  point  upon 
the  mark  using  the  upper  clamp  and  tangent  screw;  unclamp 
the  lower  motion  and  point  upon  the  star,  using  the  lower 
clamp  and  tangent  screw  and  taking  care  to  note  the  chro- 
nometer time  of  bisection ;  loosen  the  upper  motion  and  point 
again  upon  the  mark,  using  the  upper  clamp  and  tangent 
screws;  take  another  pointing  upon  the  star  with  the  lower 
motion,  noting  the  time;  point  again  upon  the  mark,  using 
the  upper  motion;  read  the  horizontal  circle.  This  completes 
the  observations  of  a  half-set  if  three  repetitions  are  to  be 
made.  In  passing  from  the  star  to  the  mark,  and  vice  versa, 
the  instrument  should  always  be  rotated  in  a  clockwise  direc- 
tion, and  the  precaution  stated  in  the  preceding  paragraph  as 
to  the  use  of  the  tangent  screws  must  be  kept  in  mind. 
Before  commencing  the  second  half-set  the  lower  motion 
should  be  undamped,  the  telescope  reversed  in  altitude,  and 
the  instrument  reversed  180°  in  azimuth.  The  program  will 
be  as  for  the  first  half-set,  except  that  now  the  first  pointing 
is  to  be  upon  the  mark;  #// pointings  on  the  mark  are  to  be 
made  with  the  lower  clamp  and  tangent  screw,  and  upon  the 
star  with  the  upper  clamp  and  tangent  screw ;  and  the  strid- 
ing level  is  to  be  read  just  after  the  last  pointing  upon  the 
star.  The  direction  of  motion  of  the  instrument  must  always 
be  clockwise  as  before,  and  the  tangent  screws  must  be  used 
as  before. 


222 


GEODETIC  ASTRONOMY. 


204. 


203.  EXAMPLE  OF   RECORD;    METHOD    OF   REPETITIONS. 

Station — Dollar  Point,  Texas.     Observer— A.  F.  Y. 

cf)  =  29°  26'  02". 6.  Instrument — Gambey  Theodolite. 

Date— April  5,  1848.  I  div.  of  striding  level  =  3". 68. 

Star — Polaris.  Chronometer — Hardy  No.  50  (Sidereal). 

a  =  ih  04™  04". 7.  Chronometer  correction  =  —  I8.8. 

d  =  88°  29'  57".82. 


Object. 

Pos. 
of 
Tel. 

Level 
Readings. 

No.  of 
Repeti- 
tions. 

Chronometer 
Times. 

Circle  Readings. 

W. 

E. 

Vernier  A. 

Vernier  B. 

Star. 

D. 

I2Q.O 

7r-5 

3 

9b  °3m33'-5 

91°  10'   30" 

271°  10'   40" 

8l.O 

119.0 

04     47  .5 

06    07  .5 

Mark. 

D. 

128     14    50 

308     14    50 

Mark. 

R. 

3 

128     14     50 

308     14     50 

9    08    06  .  5 

I2I-5 

79.0 

09     24  .5 

Star. 

R. 

80.0 

I2O.O 

10    23  .5 

91     13    40 

271     11    50 

204.  No  reading  of  the  altitude  was  taken.  The  altitude 
may  be  derived  with  sufficient  accuracy  for  use  in  computing 
the  level  corrections  from  the  table  in  §  310.  The  level  cor- 
rection must  here  be  applied  to  the  angle  between  the  star  and 
mark,  not  directly  to- the  circle  reading.  Formulae  (91)  and 
(92)  will  not  give  the  sign  of  the  level  correction;  that  must 
be  derived  from  the  consideration  that  the  star  appears  to  be 
farther  west  than  it  really  is  if  the  west  end  of  the  horizontal 
axis  is  too  high,  and  vice  versa. 

The  angle  between  the  star  and  mark,  computed  from  the 
first  half-set,  is 

(128°  14'  50"- 9'°  10' 35")*  =  I2°  2I/  25"-0, 
and  from  the  second  half-set  is 

(128°  14'  50"-  91°  13'  45")*  =  12°  20'  21".;.* 

*  Evidently  this  method  of  computing  the  second  value  of  the  angle 
necessarily  always  gives  the  same  numerical  result  as  first  computing  the 
explement  and  then  subtracting  from  360°. 


§  206.  MICROMETRIC  METHOD.  22$ 

The  remainder  of  the  computation  may  be  made  as  indi- 
cated in  §  200. 


Directions  -for  Observing  Azimuth  with  a  Micrometer. 

205.  If  the  instrument  is  provided  with  a  good  eyepiece 
micrometer  measuring  angles   in   the   plane  defined   by  the 
telescope  and  its  horizontal  axis,  the  most  accurate  as  well  as 
the  most  rapid  way  of  determining  azimuth  with  it  is  to  place 
the  azimuth  mark  nearly  in  the  vertical  plane  of  a  close  cir- 
cumpolar  star  at  elongation,  and  then  to  measure  the  hori- 
zontal angle  between  the  star  and  mark  with  the  micrometer, 
independently    of    the    graduated    horizontal    circle    of    the 
instrument. 

206.  To  place  the  azimuth  mark  with  sufficient  accuracy 
in  the  required  position  one  may  take  a  single  pointing  upon 
Polaris  on  the  first  night  after  the  station  is  ready  for  obser- 
vations, noting  the  sidereal  time  and  the  reading  of  the  hori- 
zontal circle.     The  instrument   may  then   be  left  standing, 
with  the  lower  motion  clamped,  until  the  next  day.     During 
the  next  day  the  instrument  may  be  set  to  such  a  reading  of 
the  horizontal  circle  computed  roughly  from  the  observations 
of  the  night  before,  by  the  table  of  §  310,  or  by  formula  (98), 
as  would  place  the  telescope  in  the  vertical  plane  of  the  star 
about  3Om  before  or  after  the  elongation  at  which  the  obser- 
vations are  to  be  made.     An  assistant  may  then,  by  previously 
arranged  signals,  be  aligned  at  the  proposed  site  of  the  azimuth 
mark  so  as  to  place  it  in  the  direction  defined  by  the  tele- 
scope.    The   "alignment"    of   the   mark   may   be   made   at 
night,  as  soon  as  the  pointing  is  made  upon  Polaris,  instead 
of  waiting  until  the  next  day,  if  necessary;   but  it  is  usually 
easier  to  pick  out  a  good  location  for  the  mark  and  to  trans- 
mit signals  from  the  station  to  the  mark  in  daylight  than  at 


224  GEODETIC  ASTRONOMY.  §  2O8. 

night.     The  mark  may  be  placed  either  to  the  northward  or 
to  the  southward  of  the  station. 

207.  The  adjustments  of  the  vertical  axis,  of  the  levels, 
of  focus  (see  end  of  §   216),  and  for  bringing  the  movable 
micrometer  line  into  a  vertical  plane,  must  be  made  as  indi- 
cated in  §  1 80. 

208.  The  following  is  a  good  program  for  the  observa- 
tions.     Place  the  micrometer  line  at  such  a  reading  that  it  is 
nearly  in  the   line  of  collimation  of  the  telescope.      If  this 
reading  is  not  already  known,  it  may  be  determined  by  taking 
the  mean  of  two  readings  upon  the  mark  with  the  micrometer, 
the  instrument  being  rigidly  clamped  in  azimuth,   and  the 
horizontal  axis  of  the  telescope  reversed  in  its  Ys  between  the 
two  readings.     Clamp  the  lower  circle,  and  point  upon  the 
mark  by  use  of  the  upper  clamp  and  tangent-screw.     Then, 
with  the  instrument  clamped  rigidly  in  azimuth,   take  five 
pointings  with   the   micrometer  upon   the  mark,    direct  the 
telescope  to  the  star;    place  the  striding   level   in   position; 
take  three  pointings  upon  the  star  with  the  micrometer,  noting 
the  chronometer  time  of  each;  read  and  reverse  the  striding 
level;    take   two   more  pointings  upon  the   star,   noting  the 
times;    read   the   striding  level.     This  completes  a  half-set. 
Reverse  the  horizontal  axis  of  the  telescope  in  its  Ys;  point 
approximately  to  the  star;  place  striding  level  in  position ;  take 
three  pointings  upon  the  star,  noting  the  chronometer  times; 
read  and  reverse  the  striding  level ;  take  two  more  pointings 
upon  the  star,  noting  the  times;  read  the  striding  level;  and 
finally,  make  five  pointings  upon  the  mark. 

Such  a  set  of  observations  may  be  made  very  quickly;  the 
effect  of  a  uniform  twisting  of  the  instrument  in  azimuth  is 
eliminated  from  the  result;  and  the  bubble  of  the  striding 
level  has  plenty  of  time  to  settle  without  delaying  the 
observer  for  that  purpose. 


§210. 


MICROMETRIC  METHOD. 


22$ 


209.  With  the  instrument  used  for  the  following  observa- 
tions, increased  readings  of  the  micrometer  correspond  to  a 
movement   of  the   line   of  sight  toward   the   east  when  the 
vertical  circle  is  to  the  east,  and  toward  the  west  if  the  ver- 
tical circle  is  to  the  west. 

210.  EXAMPLE    OF    RECORD   AND   COMPUTATION. 


Station  No.  10. 

0  =  31°  19'  35".o. 
Date — October  13,  1892. 
Star — Polaris,  near 

eastern  elongation. 


Observer — J.  F.  H. 

Instrument — Fauth  Theodolite  No.  725. 
One  division  of  striding  level  =  3". 68. 
Chronometer — Negus    No.    1716   (Side- 
real). 


One  turn*  of  micrometer  =  123". 73.   Chronometer  corr.  =  —  2h  nm  2S-.2. 


Cir. 
E. 
or 
W. 

Level  Readings. 

Chronometer 
Time. 

A^ 

2  sin3  *A* 

Micrometer 
Readings. 

W. 

E. 

sin  i" 

On 

Star. 

On 
Mark. 

E. 
E. 
W. 
W. 

d. 
8.0 

IO.O 

d. 
9.9 
7-3 

9h  o6m  38».o 
07     32  .0 
08    05  .5 
09     13  .0 
09     48  .0 

9     12     01  .8 

12       24   .7 
12       48   .3 

13    36  -3 

13  58  .1 

3m58'-6 
3    °4  -6 
2     31  -i 
i     23  .6 
o    48  .6 

I       25   .2 

I  48.1 

2       H    .7 

2     59  -7 
3     21  -5 

3l-°5 
18.59 
12.45 
3-82 
1.29 

3.96 
6.37 
9.46 
17.61 
22.14 

t. 
18.379 
-388 
.400 
.424 
•43° 

t. 
18.310 
•3^5 
•  315 
•3" 
.316 

Longitude 

2h  12m  west 

of  Wash- 
ington. 

Means. 
Means. 

-f  18.0 

+  0 

9.0 
7.0 

9.0 
10.9 

18.4042 

18.100 
.100 
.090 
.086 
.080 

18.0912 

18.3134 

18.290 
•275 
.279 
.281 
,279 

18.2808 

+  16.0 

-  3 

Mean  = 

-  19-9 
•9 

-  i-55 

9     10    36  .6 

12.67 

a  of  Polaris  =  ih  2om  O7".4 
d  of  Polaris  =  88°  44'  io".4 

Altitude  of  star  at  the  mean  epoch  of  the 
observations,  per  section  308,  =  31 

Altitude  of  star  at  the  middle  of  the  first 

half-set,  per  section  308,  =  31  12 

Altitude  of  star  at  the  middle  of  the 

second  half-set,  per  section  308,  =  31  14 

Collimation  reads,  ^(18.3134  -f-  18.2808)     =  18^.2971 


at  the  time  of  observation. 


13' 


*  The  head  of  this  micrometer  was  graduated  to  TOO  equal  parts, 
the  meaning  of  an  increased  reading  see  §  209. 


For 


226  GEODETIC  ASTRONOMY.  §211. 

Mark  east  of  collimation, 

18.3134  —  18.2971        =    o .0163  =  02". 02 

Circle  E.,  star  E.  of  collimation, 

(18.4042  —  18.2971)  -f-  0.8554  *  —    o  .1252 
Circle  W.,  star  E.  of  collimation, 

(18.2971  —  18.0912)  -f-  0.8551  f  =    o  .2408 

Mean,  star  E.  of  collimation  =    o  .1835  =  22  .70 


Mark  west  of  star  =  20  .68 

Level  correction,  (1.55X0.92X0.606)  =  —0.86 


Mark  west  of  star,  corrected  =  19  .82 

Mean  chronometer  time  of  observation  =    2ih   iom  36". 6 

Chronometer  correction  =  —2     n     28.2 


Mean  sidereal  time  of  observation  =     18     59     08  .4 

a  =      i     20    07 .4 


Hour-angle  (=  /)  east  of  upper  culmina- 
tion =  95°    14'    45". o  =     6     20     59  .o 
log  cos  /   =  9.9315695      Jog  sin  (f>  =  9.71593  ]og  sin  /  =  9.9981771 
log  tan  6"  =  1.6563815      log  cos  /    =  8. 96108?*  log  38.76893  =  1.5884838 

(    1.5879510  j        8.6770IW        log  tan  zn  =  8.4096933 

(  38.72140  t  —  0.04753  00  =  i°  28'    i6".92 

+  0.04753  j  /2  S}n2  i4fl  \ 

—  tan  za  -  — . \-  etc.  )  =  —  o  .33 

38.76893  n\     sini  J 

Correction  for  diurnal  aberration  =  +  o  .32 


Star  east  of  north  =  z  =  i°  28'    i6".9T 

Mark  west  of  star 

=  log  12.67  ^  1.10278        from  above  =  19  .82 

log  tan  z0          =  8.40969    Mark  east  of  north  =   i    27     57  .09 

9-51247 

o"-33 

211.  Here  again  the  sign  of  the  level  correction  as  applied 
to  the  angle  between  the  star  and  mark  must  be  derived 
directly  from  the  fact  that  the  star  appears  to  be  farther  west 

*  0.8554  =  cosine  31°  12'  (natural), 
f  0.8551  =  cosine  31°  14'  (natural). 


§21$.  MICROMETRIC  METHOD.  227 

than  it  really  is  if  the  west  end  of  the  axis  is  too  high,  and 
vice  versa. 

212.  The  micrometer  measures  angles  in  the  plane  defined 
by  the  telescope  and  its  horizontal  axis.     In  Fig.  29  let  Z  be 
the  zenith,  s  the  star  at  the  instant  when  a  pointing  is  made 
upon   it  with   the    micrometer,    and   Zn   the   vertical    circle 
described  by  the  line  of  collimation  of  the  telescope.     Then 
sm,  a  great  circle  through  s  perpendicular  to  Zn,  is  the  arc 
measured  directly  with  the  micrometer.      In  the  right  spheri- 
cal triangle  smZ  the  side  Zs  is  90°  —  A,  the  complement  of 
the  altitude  of  the  star,  and  from  Napier's  rules 

sin  sm  =  sin  mZs  cos  A .  .     .     .  t    .     (115) 

Or,  writing  the  angles  for  the  sines  of  sm  and  mZs,  and  solv- 
ing for  mZs, 

mZs  =  sm  sec  A (IJ6) 

mZs,  the  angle  at  the  zenith,  is  the  required  angle  between 
the  vertical  plane  through  the  star  and  the  vertical  plane 
described  by  the  line  of  collimation. 

The  computation  form  shows  how  this  factor,  sec  A,  is 
most  conveniently  applied.  To  be  absolutely  exact,  this 
factor  should  be  applied  to  every  pointing  upon  the  star. 
But  the  computation  as  given  is  abundantly  accurate,  the 
factor  being  applied  to  the  mean  angle  between  the  line  of 
collimation  and  the  star  for  each  half-set.  In  fact,  the  com- 
putation will  often  be  sufficiently  exact,  if  the  factor  is  applied 
to  the  mean  value  of  this  angle  for  the  set. 

213.  The  use  of  the  table  given  in  §  308  is  the  most  con- 
venient way  of  securing  the  required  values  of  the  altitude  of 
the  star,  unless  they  are  read  approximately  from  the  vertical 
circle    during  the   observations.      First   compute    the   mean 


228  GEODETIC  ASTRONOMY. 

hour-angle  of  the  star  and  take  out  the  corresponding  altitude 
for  use  in  deriving  the  level  correction,  then  the  other  two 
angles  may  be  derived  by  interpolating  over  the  interval  to 
the  middle  of  each  half-set  with  the  rate  of  change  of  altitude 
taken  from  §  308.  The  altitude  need  only  be  known  within 
one  minute,  ordinarily. 

For  any  other  star  than  Polaris,  the  table  of  §  308  not 
being  available,  one  must  either  read  the  required  altitude 
from  the  vertical  circle  of  the  instrument,  if  it  has  one,  or  else 
resort  to  the  computation  upon  which  the  table  of  §  308  is 
founded. 

The  use  of  the  factor  sec  A  is  not  necessary  with  the 
pointings  upon  the  mark,  both  because  the  line  of  collimation 
was  purposely  placed  nearly  upon  the  mark,  and  because 
sec  A  is  very  nearly  unity  for  the  small  altitude  of  the  mark. 

214.  Inspection  will  show  that  there   is  nothing  in  this 
method  of  observing  or  computing  which  limits  its  use  to  the 
time  near  elongation.     The  micrometer  may  be  used  in  this 
way  and  the  azimuth  computed  as  above  with  the  star  at  any 
hour-angle,  even  at  culmination.     But  if  the  star  is  not  near 
elongation,  its  motion  in  azimuth  is  more  rapid,  it  remains 
near  the  vertical  plane  of  the  mark  a  shorter  time,  and  larger 
angles  must   be  measured  with  the  micrometer  or  else  the 
series  of  observations  made  shorter.      Errors  in  the  time  also 
have  less  effect  the  nearer  the  star  is  to  elongation. 

If  the  azimuth  mark  is  placed  to  the  southward  of  the 
station,  the  program  of  observing  and  the  computation  are  not 
materially  modified. 

Micrometer  Value. 

215.  To  determine  the  value  of  one  turn  of  the  microm- 
eter the  observer  may  use  a  process  similar  to  that  used  in 
determining   the  value  of  the  zenith  telescope    micrometer. 


§  2 1 6.  MICROMETRIC  METHOD.  22$ 

That  is,  one  may  observe  the  times  of  transit  of  a  close  cir- 
cumpolar  star  near  culmination  across  the  micrometer  line  set 
at  successive  positions  one  turn  apart  (or  one-half  a  turn),  the 
instrument  being  rigidly  clamped  in  azimuth.  The  correc- 
tions for  curvature  may  be  made  by  use  of  the  same  table, 
§  306,  as  for  the  zenith  telescope,  but  now  using  in  the  place 
of  r  the  hour-angle  of  the  star  reckoned  from  the  nearest 
culmination,  and  making  the  corrections  to  the  observed 
times  positive  before  culmination  and  negative  after.  The 
striding  level  may  be  read  during  the  observations  and  a 
corresponding  correction  applied. 

The  correction  in  seconds  of  time  to  be  applied  to  each 
observed  time  to  reduce  it  to  what  it  would  have  been  with 
the  axis  level  is 


d.s'mA.secd       , 

-      —-     -,     (117) 


for  a  level  with  a  graduation  numbered  both  ways  from  the 
middle.  The  observer  must  depend  upon  his  instrument  to 
remain  fixed  in  azimuth, — unless,  fortunately,  he  has  an 
azimuth  mark  so  nearly  in  the  meridian  that  he  can  occa- 
sionally take  a  pointing  upon  it,  without  unclamping  the 
horizontal  circle,  during  the  progress  of  the  observations,  and 
so  determine  the  twist  of  the  instrument. 

216.  Another  convenient  way  of  determining  the  microm- 
eter value,  without  doing  any  work  at  night,  is  to  measure 
a  small  horizontal  angle  at  the  instrument  between  two 
terrestrial  objects,  both  with  the  horizontal  circle  and  the 
micrometer.  If  the  two  objects  pointed  upon  are  much  above 
or  below  the  instrument,  the  measured  angle  between  them 


230  GEODETIC  ASTRONOMY.  §  2 19. 

may  be  reduced  to  the  horizon  for  comparison  with  the  circle 
measurement  by  use  of  the  factor  sec  A,  as  indicated  in  §  2 12. 
As  the  micrometer  value  is  depended  upon  to  remain  con- 
stant for  the  station  the  focus  must  be  left  undisturbed  if 
possible  after  the  micrometer  value  has  been  determined. 

Discussion  of  Errors. 

217.  The  external  errors  are  those  due  to  errors  in  the 
right  ascension  and  declination  of  the  star  observed,  to  lateral 
refraction  of  the  rays  of  light  from  the  star  or  mark  to  the 
instrument,  and  to  error  in  the  assumed  latitude  of  the  station 
of  observation. 

218.  Errors  of  declination  enter  the  computed   azimuth 
with  full  value  when  the  star  is  observed  at  elongation,  and 
errors  of  right  ascension  enter  with  a  maximum  effect  when 
it  is  observed  at  culmination.     At  intermediate  positions  both 
errors  enter  the  computed  result  with  partial  values.     The 
errors  arising  from  this  source  are  usually  small  as  compared 
with  the  errors  of  observation,  but  are  nearly  constant  if  all 
the  observations  at  a  station  are  taken  with  the  star  at  about 
the  same  position  in  its  diurnal  path,  say  near  eastern  elonga- 
tion.    They  may  be  eliminated  to  a  considerable  extent  by 
observing  the  same  star  at  various  positions  of  its  diurnal 
path,  or  by  observing  upon  two  or  more  different  stars. 

219.  When  the  computed  results  of  a  long  series  of  accu- 
rate azimuth   observations   at   a  station   are  inspected   it   is 
usually  found  that  they  tend  to  group  themselves  by  nights. 
That  is,  the  results  for  any  one  night  agree  better  with  each 
other  than  do  the   results  on   different  nights.     They  thus 
appear  to  indicate  that  some  source  of  error  exists  which  is 
constant  during  each  night's  observations,  but  changes  from 
night  to  night.     For  example,  from  144  sets  of  micrometric 
observations  of  azimuth,  made  on  36  different  nights,  at  1 5 


§  221.  ERKOKS.  231 

stations  on  the  Mexican  Boundary  in  1892-93,  it  was  found 
that  the  error  peculiar  to  each  night  was  represented  by  the 
probable  error  ±  o".38,  and  the  probable  error  of  the  result 
from  a  single  set  exclusive  of  this  error  was  ±  o".  54.  In 
other  words,  in  this  series  of  observations,  the  error  peculiar 
to  each  night,  which  could  not  have  been  eliminated  by 
increasing  the  number  of  observations,  was  two-thirds  as 
large  on  an  average  as  the  error  of  observation  in  the  result 
from  a  single  set. 

The  most  plausible  explanation  seems  to  be  that  there  is 
lateral  refraction  between  the  mark  and  the  instrument,  and 
that  this  lateral  refraction  is  dependent  on  the  peculiar  atmos- 
pheric conditions  of  each  night.  But  whether  that  explana- 
tion be  true  or  not,  the  fact  remains  that  an  increase  of 
accuracy  in  an  azimuth  determination  at  a  given  station  may 
be  attained  much  more  readily  by  increasing  the  number  of 
nights  of  observation  than  by  increasing  the  number  of  sets 
on  each  night. 

220.  The  accuracy  with  which  the  latitude  must  be  known 
when  observing  upon  Polaris  may  be  inferred  from  an  inspec- 
tion of  the  table  of  §  310.      It  must  be  known  with  greater 
accuracy  when  a  star  farther  from  the  pole  is  used. 

221.  The  observer 's  errors  are  his  errors  of  pointing  upon 
the  mark  and  star,  errors  of  pointing  upon  the  circle  gradua- 
tion if  reading  microscopes  are  used,  errors  of  vernier  reading 
if  verniers  are  used,  errors  of  reading  the  micrometer  heads, 
errors  in  reading  the  striding  level,  and  errors  in  estimating 
the  times  of  bisection. 

There  is  such  a  large  range  of  difference  in  the  designs  of 
the  various  instruments  used  for  azimuth  work  that  little  can 
be  stated  in  regard  to  the  relative  and  absolute  magnitude  of 
these  different  errors  that  will  be  of  general  application. 
Each  observer  may  investigate  these  various  errors  for  himself 


232  GEODETIC  ASTRONOMY.  §  22$. 

with  his  own  instrument  In  designing  instruments  the 
attempt  is  often  made  to  so  fix  the  relative  power  of  the 
telescope  and  means  of  reading  the  horizontal  circle  that  the 
errors  arising  from  telescope  pointings  and  circle  readings 
shall  be  of  the  same  order  of  magnitude. 

The  effect  of  errors  in  time  may  be  estimated  by  noting 
the  rate  at  which  the  azimuth  of  the  star  was  changing  at  the 
time  it  was  being  observed.  The  table  of  §  310  will  serve 
this  purpose  for  Polaris.  Such  errors  are  usually  small,  but 
not  insensible  except  near  elongation. 

222.  Of  the  relative  magnitude  of  the  instrumental  errors 
arising  from  imperfect  adjustment  and  imperfect  construction 
little  of  general  application  can  be  said,  because  of  the  great 
variety  of  instruments  used.     With  the  more  powerful  instru- 
ments,   however,   it   may  be  stated  that   the   errors   due   to 
instability   of  the   instrument  become   relatively  great,    and 
must  be  guarded  against  by  careful  manipulation  and  rapid 
observing. 

The  errors  due  to  the  striding  level  become  more  serious 
the  farther  north  is  the  station  (see  formula  (91)).  If  the 
level  is  not  a  good  one,  it  may  be  advisable  to  take  more  level 
readings  than  have  been  suggested  in  the  preceding  programs 
of  observation.  With  a  very  poor  level,  or  at  a  station  in  a 
high  latitude,  it  may  be  well  to  avoid  placing  any  dependence 
upon  the  level  by  taking  half  of  the  observations  upon  the 
star's  image  reflected  from  the  free  surface  of  mercury  (an 
artificial  horizon).  The  effect  of  inclination  of  the  axis  upon 
the  circle  reading  will  be  the  negative  for  the  reflected  star  of 
what  it  is  for  the  star  seen  directly.  Considerable  care  will 
be  necessary  to  protect  the  mercury  from  wind  and  from 
tremors  transmitted  to  it  through  its  support. 

223.  The  micrometric  method  treated  in  §§  205-2  14  gives 
a  higher  degree  of  accuracy  than  the  other  methods  described, 


§22$.  MISCELLANEOUS.  233 

if  a  good  micrometer  is  available.  It  avoids  several  sources 
of  error,  and  the  observations  may  be  made  so  rapidly  that 
the  conditions  are  quite  favorable  for  the  elimination  of  errors 
due  to  instability.  The  error,  in  the  final  result  for  a  station, 
due  to  an  error  in  the  value  of  the  micrometer  screw  may  be 
made  as  small  as  desired  by  so  placing  the  azimuth  mark  and 
so  timing  the  observations  that  the  sum  of  the  angles  meas- 
ured eastward  from  the  mark  to  the  star  shall  be  nearly  equal 
to  the  sum  of  the  angles  measured  westward  from  the  mark 
to  the  star. 

Other  Instruments  and  Methods. 

224.  An  astronomical  transit  furnished  with  an  eyepiece 
micrometer  is  especially  well  adapted  to  give  results  of  a  high 
degree  of  accuracy  in  determining  azimuths  by  the  micrometric 
method. 

225.  If  the  transit  has  no  micrometer,  a  secondary  azimuth 
may  be  determined  incidentally  to  time   observations  with 
little  extra  expenditure  of  time.      Put  an  azimuth  mark  as 
nearly  as  possible  in  the  meridian  of  the  transit.     At  the 
beginning  of  each  half-set  of  the  time  observations  point  upon 
the  mark  with  the  middle  line  of  the  reticle.      If  the  mark  is 
nearly  in  the  horizon  of  the  instrument,  the  collimation  and 
azimuth  errors  of  the  transit  as  derived  from  each  half-set, 
reduced  to    arc  and  combined  by  addition   and  subtraction 
with  each  other  and  with  the  equatorial  interval  of  the  middle 
line,  give  the  azimuth  of  the  mark.     The  azimuth  of  a  certain 
mark  was  so  determined  from  the  time  observations  required 
for  a  determination  of  the  longitude  of  a  station  at  Anchorage 
Point,    Chilkat    Inlet,    Alaska,    in     1894.       The    computed 
azimuth  of  the  mark  from   38   nights  of  observation   varied 
through  a  range  of  12". 8.     The  probable  error  of  a  single 
determination  was  ±  2".!. 


234  GEODETIC  ASTRONOMY.  §  228. 

226.  The  transit  may  also  be  made  to   furnish   a  good 
determination  of  azimuth  by  observations  in  the  vertical  of 
Polaris  by  the  method  already  referred  to  in  §  129. 

227.  If  the  allowable  error  of  a  given  azimuth  determina- 
tion exceeds   2",   a  convenient  method  is  to   observe  upon 
Polaris  at  any  hour-angle  and  use  the  table  given  in  §  310  to 
compute  its  azimuth  at  the  time  of  each  observation.     The 
tabulated  values  *  were  computed  by  formula  (98).     The  only 
correction  to  be  applied  to  the  value  as  taken  from  the  table 
is  that  due  to  the  difference  between  the  apparent  declination 
of  Polaris  at  the  time  of  observation  and  the  value  88°  46' 
with  which  the  table  was  computed.     This  may  be  computed 
by  use  of  the  columns,  given  at  the  right-hand  side  of  the 
table,  headed  "  Correction  for   if  increase  in  declination   of 
Polaris,"  by  assuming  that  the  correction  is  proportional  to 
the  increase,  and  must  be  changed  in  sign  if  the  declination 
is  less  than  88°  46'.     The  table  may  also  be  used  as  a  con- 
venient rough  check  on  computations  made  by  formula  (98). 

If  a  star  which  is  not  a  close  circumpolar,  or  the  Sun,  is 
observed  for  azimuth  at  a  known  hour-angle,  its  azimuth  may 
be  computed  by  formula  (98)  for  each  observation,  or  the 
observations  may  be  treated  in  groups  covering  short  intervals 
of  time.  But  formula  (103)  will  not  apply,  since  certain 
approximations  were  made  in  its  derivation  which  are  only 
allowable  when  d  is  nearly  90°. 

228.  For   rough  determinations  of  azimuth   in   daylight, 
say  within  30",  when  the  time  is  only  approximately  known, 
the  Sun  may  be  observed  with  a  small  theodolite,  or  with  an 
engineer's  transit,  as  follows:    Point  upon  the  mark  and  read 
the  horizontal  circle;   point  upon  the  Sun,  making  the  hori- 
zontal line  of  the  transit  tangent  to  the  upper  limb  and  the 

*  See  Coast  and  Geodetic  Survey  Report,  1895,  Appendix  No.  10,  for  the 
original  of  this  and  the  following  table. 


§  228.  MISCELLANEOUS. 

vertical  line  tangent  to  the  western  limb;  note  the  time,  and 
read  both  horizontal  and  vertical  circles;  repeat  this  pointing 
upon  the  Sun  twice  more,  noting  the  times  and  reading  the 
circles;  reverse  the  instrument  180°  in  azimuth  and  the  tele- 
scope in  altitude;  again  take  three  readings  upon  the  Sun, 
but  now  make  the  horizontal  line  tangent  to  the  lower  limb 
and  the  vertical  line  tangent  to  the  eastern  limb ;  finally,  point 
upon  the  mark  again  and  read. 

To  compute  the  azimuth  of  the  Sun  one  may  use  the 
formula 

sin  (s  —  0)  sin  (s  —  A) 

tan2  4*  =  -  /         px      i       •     •       1 1 8) 

cos  s  cos  (s  —  P) 

in  which  z  is  the  azimuth  counted  from  the  north,  P  is  the 
Sun's  north  polar  distance  (=  90°  —  #),  and  s  =  4(0  -|~  A 

+f)' 

If  desired,  the  hour-angle  of  the  Sun,  and  thence  the 
chronometer  error,  may  also  be  computed  from  the  observa- 
tions by  the  formula 

cos  s  sin  (s  —  A) 

tan   4*  =  - — 7 -7T-1 — 7 — £-=.      .     .     (i  19) 

sin  (s  —  0)  cos  (s  —  P) 

These  formulae  may  readily  be  derived  from  the  ordinary 
formulae  of  spherical  trigonometry  as  applied  to  the  triangle 
defined  by  the  Sun,  the  zenith,  and  the  pole. 


236 


GEODETIC  ASTRONOMY. 


§229. 


229. 


Observations  of  Sun  for  Azimuth, 


Niantilik,  Cumberland  Sound,  British  America,  Sept.  18,  1896,  P.M. 
Instrument — Theodolite  Magnetometer  No.  19.     <f>  =  64°  53'. 5. 
Chronometer  correction  on  Greenwich  Mean  Time  -f  2m  og'.S. 


Object. 

c 

Time. 
Chronome- 
ter, 1842. 

Horizontal  Circle. 

Vertical  Circle. 

A. 

B. 

Mean. 

A. 

B. 

Mean. 

D. 
R. 

R. 
D. 

D. 
R. 

R. 
D. 

7h  39m  20" 
40    40 
41     34 

42     58 
44     04 
45     33 

7     42     21   .5 

7     46    34 
47     33 
48     31 

51     04 
S2     05 
53     21 

53°  58' 
233     55 

53 

229     46 
230    05 
19 

51     16 

33 
54 

50 
S2    09 

11 

232     38 
52 
233     I2 

59' 
55 

44 

°1 
16 

17 
33 
55 

09 
24 

38 

36 
50 

09 

55 
58 

58'.  s 
55  -° 

"56.8 

45  -o 
04  .0 
17  -5 

16.5 
33  -o 
54  -5 

~&~ 

og  .0 
23  -5 
38  .0 

37  -o 
51  .0 
10  .5 

17°  17' 
ii 

06 

73    33 

32 

46 

16 

73     50 
55 
74    02 

16     16 
ii 
03 

18' 
13 

09 

33 
39 
47 

57 
03 

17 
07 

17'.  5 

12   .0 

07  -5 

33  -o 
39  .0 
40  .5 

46.4 

50.5 
56  .0 

02    .5 

I6.5 
12   .0 

05  .0 

Sun's  first  and  upper  limb.. 
Sun's  second  and  lower  limb 
Means  

Sun's  second  and  lower  limb 
Sun's  first  and  upper  limb.  .  . 

7     49     5i   -3 

52 

233     56 

53     58 

53 

38.2 

55  -5 
58.0 

56  .8 

16 

07   .4 

Azimuth  mark.  .  .  .          ... 

Means               .     . 

§230. 


MISCELLANEO  US. 

Computation. 


237 


Chronometer  time  *  .......................... 

Chronometer  corr.  on  Greenwich  Mean  Time.. 
Greenwich  Mean  Time  ....................... 

Sun's  Apparent  Declination,  <5,  interpolated 
from  Ephemeris  ..................... 

Observed  Altitude  ..................... 

Correction  for  parallax  ................ 

Correction  for  refraction  ............... 

Corrected  Altitude,  A  ................. 

P(=go°-d)  .......................... 

Latitude,  <p  ............................ 


s-  A 

s-P 

Log  sin  (s  —  <p) 

Log  sin  (s  —  A) 

Log  numerator 

Log  cos  s 

Log  cos  (s  —  P) 

Log  denominator 

Log  tan*  \z 

Log  tan  \z 


Horizontal  circle  reads. 
True  meridian  reads... 
Azimuth  mark  reads... 
Mark  west  of  north. . . . 


First  Half. 

Second  Half. 

7h  42m  2I8.5 

7h  49m  5  1".  3 

-f-  02  09  .8 

+  02   09.8 

7  44  3i  -3 

7  52  01.  i 

i°  26'.  9 

i°  26'.  8 

16  46  .4 

16  07  .4 

+  0.1 

+  0.1 

-  3-1 

-  3-3 

16  43.4 

16  04.2 

88  33.1 

88  33.2 

64  53-5 

64  53  -5 

85  05  .0 

84  45  .4 

20   II.5 

19  51-9 

68  21  .6 

68  41  .2 

—  3  28.1 

-3  47-8 

9.53802 

9.53124 

9.96826 

9.96923 

9.50628 

9.50047 

8.93301 

8.96090 

9.99920 

9.99904 

8.93221 

8.95994 

0.57407 

0.54053 

0.28704 

0.27026 

62°  41'.  4 

61°  46'.  6 

125   22.8 

123  33  .2 

50  48  .4 

52  38.2 

176   II  .2 

176  ii  .4 

53  56.8 

53  56.8 

122   14  .4 

122   14  .6 

230.  The  derived  value  of  the  azimuth  is  more  exact  the 
farther  the  Sun  is  from  the  meridian,  and  becomes  unreliable 
when  the  Sun  is  very  near  the  meridian.  These  are  the  same 
conditions  that  limit  the  use  of  the  solar  transit.  If,  how- 
ever, the  error  of  the  timepiece  has  been  determined  earlier 
in  the  day  by  formula  (i  19),  or  is  determined  later  in  the  day, 


*  It  is  important  to  notice  that  the  only  way  in  which  this  observed 
chronometer  time  enters  this  computation  is  as  a  means  of  interpolating  the 
declination  from  the  Ephemeris. 


238  GEODETIC  ASTRONOMY.  §  231. 

or  both,  an  observation  at  noon  will  give  a  good  determina- 
tion of  azimuth  by  computing  the  hour-angle  from  the 
observed  times  and  then  computing  the  azimuth  from  the 
formula 

cos  (s  +  d  —  90°) 

,     •     •     (120) 


which  may  be  derived  readily  from  (118)  and  (119). 


QUESTIONS   AND    EXAMPLES. 

231.  i.  In  the  azimuth  computation  of  §  200  a  correction 
is  applied  for  diurnal  aberration.  Why  is  not  a  correction 
also  applied  for  the  aberration  due  to  the  motion  of  the 
Earth  in  its  orbit  ? 

2.  Differentiate  equation  (98)  with  respect  to  0,  and  show 
in  a  general  way  the  relative  errors  introduced,   by  a  given 
error  in   0,   into  the  computed    azimuth   from  observations 
taken  upon  stars  of  various  declinations  observed  at  various 
hour-angles    and    at    stations    in    various    latitudes.       Check 
your  conclusions  by  inspection  of  the  table  in  §  310.     Also 
do  the  same  with  respect  to  errors  of  time. 

3.  To  what  indeterminate  form  does  formula  (98)  reduce 
for  a  star  in  the  zenith  ?     Is  that  case  in  nature  indetermi- 
nate ? 

4.  What   difficulties  would   you   expect  to  encounter  in 
determining  the  azimuth  accurately  at  a  station  of  which  the 
latitude   is   nearly   90°,    aside   from   those   arising   from    the 
climate  ?    In  considering  this  question  remember  that  in  most 
of  the  methods  for  determining  the  azimuth  the  determination 
of  the  error  of  a  chronometer  on  local  time  is  one  of  the 
necessary  auxiliary  observations. 


§231. 


QUESTIONS  AND   EXAMPLES. 


239 


5.  What  is  the  azimuth  of  the  mark  from  the  record  given 
in  §  203  ? 

6.  What  is  the  azimuth  of  the  mark  from  the  following 
record  ? 

Station  No.  14.  February  u,  1893. 

Observations  for  azimuth  oi  mark  on  Polaris  near  western  elongation. 
Chronometer  error  =  +  2h  29W  478«2-     One  division  of  level  =  3". 68. 
One  turn  of  micrometer  =  123". 73.         0  =  32°  29'  oi".  12. 


Cir. 
E. 

Level  Readings. 

Chronometer 

Micrometer 
Readings. 

or 

Time. 

W. 

W. 

E. 

On  Star. 

On  Mark. 

d 

d 

t 

t 

E. 

7.2 

6-3 

8"  47m488.5 

19.219 

18.400 

Longitude 

7.2 

6.4 

48     31    o 

.189 

•391 

2h  3Im  WeSt 

49    oi    5 

.170 

•387 

of      Wash- 

+ 14-4 

—  12.7 

49     47    o 

.146 

•399 

ington. 

E. 

50    20    o 

.124 

•393 

19.1696 

18.3940 

Means. 

W. 

6  o 

8.0 

8    52    09    o 

17.790 

18.470 

6.1 

7-8 

52     36    5 

.800 

.470 



53    04    5 

.820 

.469 

+  12.  I 

-isT 

53     57    5 

.8S2 

•475 

W. 

—      2.0 

=  Sum 

54     28    o 

.871 

.462 

17.8266 

18.4692 

Means. 

a  of  Polaris 


$  of  Polaris 


44'  33"-4 


7.  Prove  formula  (117)  for  the  level  correction  to  be  made 
when    determining    the    value    of    an    eyepiece    micrometer 
measuring  angles  in  the  plane  of  the  telescope  and  its  hori- 
zontal axis. 

8.  Prove  the  statement  of  §  225,  that  under  certain  con- 
ditions the  azimuth  error,   collimation  error,  and  equatorial 
interval  of  the  middle  line  of  a  transit  when  combined  by 
simple  addition  and  subtraction  give  the  azimuth  of  the  mark. 

9.  What  is  the  circle  reading  corresponding  to  the  true 
north  from  the  following  record  ? 


240 


GEODETIC  ASTRONOMY. 


§231 


Station — Capitol,  East  Park,  Washington,  D.  C. 

Sun  near  prime  vertical,  August  15  A.M.,  1856.     Observer — C.  A.  S. 

Instrument — 5-in.  Magnetic  Theodolite.     Sidereal  Chronometer. 


Chronometer 
Time. 

Horizontal  Circle. 

Vertical  Circle. 

78°  Fahr. 

A 

B 

A 

B 

Sun's  upper  and  first  limb.    Telescope  D. 

5h  20m  44.0s 
22  01.5 

25  26.5 

5h  27">32.5' 
28  39.5 
30  oi.o 

28°  25'  oo" 
28   37  45 
29    13  30 

208°  25'  oo" 
208  38  15 
209   14  oo 

58°  29'  oo" 
58    14  45 
57   36  oo 

58°  29'  30" 
58    14  30 

57    35  45 

Sun's  lower  and  second  limb.     Telescope  R. 

209°  oi  '  30" 
209   12  45 
209  27  oo 

29°  oo'  30  ' 

29     12    15 
29    26    30 

57°  48  oo" 
57   34  30 
57    19  15 

57°  47'  30" 
57   34  15 
57    18  30 

0  —  38°  53'  18"     A  =  5h  o8m  oi'.o  west  of  Greenwich. 

6  (at  mean  of  the  times)  =  13°  55'  16".     (Interpolated  from  Ephemeris.) 

10.  If  both  are  available,  which  should  be  used  in  formula 
(98) — the  geodetic  or  the  astronomical  latitude  ? 

11.  Prove  formulae  (118),  (119),  and  (120). 


§  233-  LONGITUDE.  241 


CHAPTER  VII. 
LONGITUDE. 

To  determine  the  longitude  of  a  station  on  the 
Earth's  surface,  referred  to  the  meridian  of  Greenwich,  is  to 
determine  the  angle  between  the  two  meridian  planes  passing 
through  the  station  and  Greenwich  respectively.  (See  §  15.) 
This  angle  between  the  two  meridian  planes  is  the  same  as 
the  difference  of  the  local  times*  of  the  two  stations,  con- 
sidering 24h  to  represent  360°.  (See  §  21.)  Hence  to  deter- 
mine the  longitude  of  a  station  is  to  determine  the  difference 
between  the  local  time  of  that  station  and  the  local  time  of 
Greenwich.  In  general  the  longitude  of  an  unknown  station 
is  not  referred  to  Greenwich  directly,  but  to  some  station  of 
which  the  longitude  is  already  known.  The  astronomical 
determination  of  the  longitude  of  a  station  consists,  then,  in 
a  determination  of  the  local  time  at  each  of  two  stations,  the 
longitude  of  one  which  is  known  and  of  the  other  is  to  be 
determined,  and  the  comparison  of  these  two  times.  Their 
difference  is  the  difference  of  longitude  expressed  in  time. 
This  may  be  reduced  to  arc  by  the  relations  24**  =  360°, 
I*  =  15°,  im  =  15',  and  I8  =  15". 

233.   The  principal  methods  of  determining  differences  of 

*  The  times  may  be  eifher  sidereal  or  mean  solar.  The  vernal  equinox 
apparently  makes  one  complete  revolution  about  the  earth  in  24  sidereal 
hours,  and  the  mean  Sun  apparently  makes  one  complete  revolution  in  24 

mean  solar  hours. 


242  GEODETIC  ASTRONOMY.  §  235. 

longitude  are  by  the  use  of  the  telegraph,  by  transportation 
of  chronometers,  by  observations  of  the  Moon's  place,  and 
by  observations  of  eclipses  of  Jupiter's  satellites.  The 
methods  of  making  the  necessary  determinations  of  the  local 
time  in  each  of  these  methods  need  not  be  considered  here, 
as  they  have  already  been  exploited  in  Chapters  III  and  IV. 
We  need  here  consider  only  the  methods  by  which  some 
signal  is  transmitted  between  the  stations  to  serve  for  the 
comparison  of  the  times. 

The  Observing  Program  and  Apparatus  of  the  Telegraphic 

Method. 

234.  The  telegraphic  method  has  been  used  very  exten- 
sively in  this  country  by  the  Coast  and  Geodetic  Survey,  and 
during  the  fifty  years  of  its  use  has  been  gradually  modified. 
The   method   and   apparatus    at   present   used    will    be   here 
described. 

The  nightly  program  at  each  station  is  to  observe  two  sets 
of  ten  stars  each  for  time  with  a  transit  of  the  type  shown  in 
Fig.  10.  Each  half-set  consists  in  general  of  four  stars 
having  a  mean  azimuth  factor  A  (see  §  299)  nearly  equal  to 
o,  and  one  slow  star  (of  large  declination)  observed  above 
the  pole.  Two  such  half-sets,  with  a  reversal  of  the  telescope 
in  the  Ys  between  them,  give  a  strong  determination  of  the 
time.  The  same  sets  of  stars  are  by  previous  agreement 
observed  at  each  station.  Between  the  two  time  sets,  or 
rather  at  about  the  middle  of  the  night's  observations,  certain 
arbitrary  signals  are  exchanged  by  telegraph  between  the  two 
stations,  which  serve  to  compare  the  two  chronometers,  and 
therefore  to  compare  the  two  local  times  which  have  been 
determined  from  the  star  observations.* 

235.  Fig.    30  shows   the  arrangement   of    the    electrical 
apparatus  at  each  station  during  the  intervals  when  no  arbi- 


§236.     TELEGRAPHIC  LONGITUDE    OBSERVATIONS.  243 

trary  signals  are  being  sent  or  received,  and  each  observer  is 
busy  taking  his  time  observations.  In  the  local  circuit,  which 
is  now  entirely  independent  of  the  Western  Union  lines,  are 
placed  the  break-circuit  chronometer*  (or  clock),  battery, 
chronograph,  and  the  break-circuit  observing  keys.  All  the 
time  observations  are  recorded  on  the  chronograph.  Mean- 
while the  telegraph  operator  has  at  his  disposal  the  usual 
telegrapher's  apparatus  upon  the  main  line  connecting  the  two 
stations,  namely,  his  key,  and  the  sounder  relay  which  con- 
trols the  sounder  in  a  second  local  circuit.  The  operator,  a 
few  minutes  before  the  time  for  exchange  of  signals,  secures 
a  clear  line  between  stations,  ascertains  whether  the  observa- 
tions at  the  other  station  are  proceeding  successfully,  and 
finally  an  agreement  is  telegraphed  between  the  two  observers 
as  to  the  exact  epoch  at  which  the  exchange  of  signals  will  be 
made. 

236.  When  that  epoch  arrives,  time  observations  are 
stopped  at  each  station,  and  by  suitable  switches  the  electrical 
apparatus  at  each  station  is  arranged  as  shown  in  Fig.  31. 
The  only  change  is  that  now  a  relay,  called  a  signal  relay,  is 
used  to  connect  each  local  chronograph  circuit  with  the  main 
line  in  such  a  way  that  the  local  circuit  will  be  broken  every 
time  the  main  circuit  is  broken,  in  addition  to  the  regular 
breaks  made  in  it  by  the  local  chronometer.  The  observer 
at  station  A  now  takes  the  telegrapher's  key  (in  the  main  cir- 
cuit), and  sends  a  series  of  arbitrary  break-circuit  signals  over 
the  main  line  by  holding  the  key  down  except  when  a  dot  is 
sent  by  releasing  the  key  for  an  instant — the  reverse  of  the 
ordinary  usage  of  the  telegrapher.  He  listens  to  his  own 
chronograph,  and  sends  a  signal  once  in  each  two-second 
interval  at  such  an  instant  as  will  not  conflict  with  his  own 

*  Or  the  chronometer  maybe  placed  in  a  separate  local  circuit,  breaking 
this  one  through  a  relay. 


244  GEODETIC  ASTRONOMY.  §  237« 

chronograph  record.  Each  signal  is  transmitted  by  the 
signal  relay  at  station  A  to  that  local  circuit,  and  its  time  of 
receipt  recorded  automatically  by  the  chronograph.  At  the 
same  instant,  except  for  the  time  required  for  the  electrical 
wave  to  be  transmitted  over  the  main  line  between  stations, 
the  signal  relay  at  station  B  transmits  the  signal  to  the  local 
circuit  and  chronograph  there.  If  these  signals  coincide  with 
the  clock  breaks  on  the  chronograph  at  B  at  any  time,  the 
observer  at  B  breaks  into  the  main  circuit  with  his  teleg- 
rapher's key,  and  produces  a  rattle  at  A's  sounder  which 
informs  him  that  he  must  change  his  signals  a  fraction  of  a 
second  to  another  part  of  the  intervals  given  him  by  his 
chronograph  beat.  Thirty  signals  are  sent  from  station  A  at 
intervals  of  about  two  seconds.  The  observer  at  A  then 
closes  his  key  and  the  observer  at  B  proceeds  to  send  thirty 
similar  signals  from  B  to  A.*  The  Western  Union  line  is 
then  released,  the  apparatus  at  each  station  is  again  arranged 
as  shown  in  Fig.  30,  and  each  observer  proceeds  to  finish  his 
time  observations. 

For  a  first-class  determination  this  program  is  carried  out 
for  five  nights  at  each  station;  the  observers  then  change 
places  (to  eliminate  the  effect  of  personal  equation),  the 
instrumental  equipment  of  each  station  being  left  undisturbed ; 
and  the  same  program  is  again  followed  for  five  nights. 

Example  of  Computation. 

237.  A  determination  of  the  difference  of  longitude  of 
Cambridge,  Mass.,  and  of  Ithaca,  N.  Y.,  was  made  May  16- 
June  3,  1896.  The  following  is  a  portion  of  the  field  com- 
putation: 

*  Thirty  signals  at  two-second  intervals  keep  each  chronometer  in  use 
timing  signals  for  just  one  revolution  (im)  of  the  toothed  wheel  which 
breaks  the  circuit  in  the  chronometer,  and  thus  any  errors  in  the  spacing  on 
that  wheel  are  eliminated  from  the  final  result. 


238.    COMPUTATION  OF  TELEGRAPHIC  LONGITUDES.     24$ 
Arbitrary  Signals,  May  27,  1896. 


From  Ithaca  to  Cambridge. 

From  Cambridge  to  Ithaca. 

Cambridge  Record. 

Ithaca  Record. 

Cambridge  Record. 

Ithaca  Record. 

I4h    l6m  46s.  34 

48.39 
50.32 

52.48 

9h   38™  19".  52 
21  .54 
23.50 
25.63 

I4h  I7m  569-55 
58.51 
oo  .56 

02  .50 

gh  39m  29s.  63 
31  .61 
33.63 
35-57 

42.41 
44.31 
46.35 

48.52 

15  -45 
17.30 
19.36 

21   .52 

50.37 
52.40 

54  -44 
56.45 

23-3I 
25  .36 
27-39 
29-39 

I4h  i7m  i78-44i 
-  25  .702 

14     16     51  .739 
Difference  ; 

9h  38'"  50'.  532 
-07     57-107 
9     30     53  .425 
i     33.783 

4      22       59.370 

13     55     26.578 
Jim  25*.i6i 

I4h   i8m  26».  426 
-  25  .700 

14     18     00.726 
Difference  : 

9h   39m  59M40 
—  07     57  .109 
9     32     02  .331 

i     33.971 
4     22     59  .370 
13     56     35  -672 

2im  25*.054 

The  heading  shows  which  way  the  signals  were  sent  over 
the  main  line,  and  the  four  columns  give  the  times  of  the 
signals  as  read  directly  from  the  chronograph  sheets  at  the 
stations  indicated.  There  were  31  or  32  signals  in  each  series, 
of  which  only  a  portion  are  here  printed.  The  means  are 
given  for  the  whole  series  in  each  case.  A  mean-time  clock 
was  used  in  the  chronograph  circuit  at  Ithaca,  and  a  sidereal 
chronometer  at  Cambridge. 

238.  The  first  time  set  of  the  evening  at  Cambridge  gave 
for  the  chronometer  correction,  on  local  sidereal  time,  at  the 
mean  epoch  of  the  set,  when  the  chronometer  read  I3h  3Om.2, 
—  2  5s. 787.  The  second  set  gave  the  chronometer  correction 
=  —  2  5 ".677  at  the  epoch  when  the  chronometer  read  I4h 
3im.3.  By  taking  the  means  of  the  epochs  and  corrections, 


246  GEODETIC  ASTRONOMY.  §  239. 

on  the  assumption  that  the  chronometer  rate  was  constant 
during  this  interval,  it  was  found  that  the  correction  was 
—  2  5s.  732  at  the  chronometer  reading  I4h  oo"3./.  Also  from 
the  differences  of  epochs  and  corrections  it  was  found  that  the 
rate  of  the  chronometer  during  this  interval  was  o8.  00180  per 
minute.  Applying  this  rate  for  the  interval  (i4h  17™.  3  —  I4h 
oom.7)  to  the  value  —  2  5s.  732  of  the  correction,  there  is 
obtained  for  the  chronometer  correction  at  the  mean  epoch 
of  the  signals  sent  from  Ithaca  to  Cambridge  —  2  5s.  702. 
Similarly,  from  the  time  observations  at  Ithaca  it  was  found 
that  when  the  clock  read  Qh  38m.8  its  correction  was  —  7m 
57s.  107  (on  local  mean  solar  time).  The  computation  *  shows 
how  the  mean  epoch  of  the  signals  was  derived  in  Cambridge 
sidereal  time  (i4h  i6m  5  Is.  739),  and  in  Ithaca  sidereal  time 
(i3h  55m  268.578)f.  The  difference  of  these  two,  2im  258.i6i 
is  the  difference  of  longitude  of  the  stations,  affected  by  the 
transmission  time  of  the  electric  wave,  and  by  the  relative 
personal  equation  of  the  two  observers. 

239.  The  longitude  difference  as  computed  from  the  other 
set  of  signals  shown  in  the  computation  is  evidently  affected 
in  the  reverse  way  by  the  transmission  time.  Hence  the 
mean  of  the  two  derived  values,  namely,  £(2im  25s.  161  -f-  2im 
258-°54)  =  2im  25s.  108,  is  the  longitude  difference  unaffected 
by  transmission  time,  provided  such  time  remained  constant 
during  the  two  minutes  of  the  exchange.  Also,  the  transmis- 
sion time:);  itself  is  J(2im  25s.i6i—  2im  25".O54)  = 


*  This  computation  would  be  simplified  in  an  obvious  manner  if  sidereal 
timepieces  had  been  used  at  both  stations. 

f  4h  22™  59".  370  is  the  sidereal  time  of  mean  moon  at  Ithaca  May  27, 
1896. 

\  The  mean  value  of  the  transmission  time  on  nine  nights  over  this  line 
was  0.070,  and  the  separate  values  varied  from  O".O54  to  o'.o84.  The  tele- 
graph line  from  Ithaca  to  Cambridge,  by  way  of  Syracuse,  New  York,  and 
Boston,  was  592  miles  long,  and  passed  through  one  repeater  (at  New  York). 


§240.    COMPUTATION  OF  TELEGRAPHIC  LONGITUDES.     247 

An  inspection  of  Fig.  31  will  show  that  this  is  merely  the 
transmission  time  between  the  two  signal  relays,  and  does  not 
include  the  transmission  time  through  the  relays  and  the 
chronograph  circuit,  as  this  part  of  the  transmission  is  always 
in  one  direction,  no  matter  where  the  signal  starts  from  in  the 
main  circuit. 

240.  In  the  regular  program  of  observation  *  five  values 
of  the  longitude  would  thus  be  obtained,  and  then  five  more 
similar  results  after  the  observers  exchanged  places.  From 
these  two  means  the  effect  of  relative  personal  equation  would 
then  have  been  eliminated  by  computation,  as  shown  in  the 
portion  of  a  field  computation  given  below. 

DIFFERENCE  OF  LONGITUDE. 
Albany,  N.  Y.,  west  of  Montreal. 


Date,  1896. 
Sept.  16 

"         20 

"     24  .     . 
"      28      . 
Oct.  9...     . 

Oct.  10.      . 
"     I5-- 
'     19.. 

'*       21  .. 
"       26..        . 

Observer 
at 

Longitude 
Difference. 
A  Signals. 

Longitude 
Difference 
M  Signals. 

A-M 

0.044 

0.039 
0.038 
0.035 
0.036 

Mean  of 
A  and  M 
Signals. 

d*  4i».o28 
41  .066 
41  .017 
4»  -005 
41  .034 

Per- 
sonal 
Equa- 
tion. 

+  0.268 
-0.268 

Difference 
of 
Longitude. 

A 

M 

F 
F 
F 
F 
F 

S 
S 
S 

s 

s 

S 
S 

s 
s 

s 

F 
F 
F 
F 
F 

om  4i'-°5° 
41  .086 
41  .036 
41  .022 
41  .052 

o    41  .525 
41  -569 
41  .617 
41  -639 
4i  -578 

om  4i«.oo6 
41  .047 

40  .998 
40  .987 
41  .016 

Means  — 

o    41  .491 
41  -538 
4i  .580 
41  -599 
41  .526 

Means  = 

Om  4I«.296 

4i  -334 
41  -285 
4i  -273 
41  .302 

41  .240 
41  .285 
4i  -330 
4i  -351 
41  .284 

0.038 

0.034 
0.031 
0.037 
0.040 
0.052 

o    41  .030 

o    41  .508 
41  -553 
41  .598 
41  .619 
4i  -552 

0.039 

o     41  .566 

o    41  .298 

Transmission  time  =  K°"-038)  =  o8.oig. 

Relative  personal  equation,  S  —  F  =  $(41.566  —  41.030)  =  -(-  o§.268. 

Difference  of  longitude,  A  —  M  =  oh  oom  41*. 298. 


*  The  regular  program  was  not  carried  out  at  this  station, — hence  the 
following  illustration  is  taken  from  another  source. 


248  GEODETIC  ASTRONOMY.  §  242. 

Discussion  of  Errors. 

241.  From  the  final  computed  result  there  has  thus  been 
eliminated  the  average  relative  personal  equation  during  the 
series  of  observations,  and  the  average  value  of  the  transmis- 
sion time  during  the  short  interval  covered  by  the  exchange 
of  signals  on  each  evening.      The  errors  of  the  adopted  right 
ascensions  are  also  eliminated  from  the  result,   because  the 
same  stars  have  been  observed  at  both  stations.* 

242.  The  final  computed  result  is  subject  to  the  following 
errors:    1st,  that  arising  from  the  accidental  errors  of  observa- 
tions of  200  stars  at  each  station,  which  must  be  quite  small 
after  the  elimination  due  to  400  repetitions;  2d,  that  arising 
from  the  variation  of  the  relative  personal  equation  of  the  two 
observers  from  night  to  night,  of  which  the  magnitude  may 
be  estimated  from  the  following  paragraphs;   3d,  that  due  to 
lateral  refraction,  to  which  reference  will  be  made  in  §  245 ; 
4th,  that  due  to  variations  in  the  rates  of  the  chronometers 
during  the  period  covered  by  the  observations,  which  must 
usually  be  quite  small,  as  the  chronometers  are  not  disturbed 
in  any  way  during  the  observations  and  are  protected  as  far 
as  possible  against  changes  of  temperature;   5th,  that  arising 
from  the  variation  of  the  transmission  time,  between  the  two 
halves  of  the  exchange  of  signals,   on  each  night,  which  is 
probably  insensible,  as  this  interval  is  usually  only  a  minute ; 
6th,    the   difference   of  transmission   time   through    the   two 
signal  relays,  since  this  difference  always  enters  with  the  same 
sign,  as  may  be  seen  by  an  inspection  of  Fig.  31.     This  last 
error  is  made  very  small  by  using  specially  designed  relays 
which  act  very  quickly,  by  adjusting  the  two  relays  to  be  as 

*  Where  the  difference  of  longitude  is  very  large,  the  observers  may  be 
forced  to  use  different  star  lists  to  avoid  depending  upon  their  chronometer 
rates  for  too  long  an  interval. 


§243-  PERSONAL   EQUATION1.  249 

nearly  alike  as  possible,  by  controlling  the  strength  of  the 
current  passing  through  the  relay  so  that  it  shall  always  be 
nearly  the  same,  and  by  exchanging  relays  when  the  observers 
change  places,  or  by  a  combination  of  these  methods.* 

Personal  Equation. 

243.  The  extent  to  which  the  relative  personal  equation 
may  be  expected  to  vary  may  be  estimated  from  the  follow- 
ing statement  of  the  experience  of  two  observers  who  have 
made  the  major  portion  of  the  primary  longitude  determina- 
tions of  the  Coast  and  Geodetic  Survey  during  the  period 
indicated.  The  plus  sign  indicates  that  Mr.  Sinclair  observes 
later  than  Mr.  Putnam. 

PERSONAL  EQUATION  BETWEEN  C.  H.  SINCLAIR  AND  G.  R. 
PUTNAM,  ASSISTANTS  C.  AND  G.  SURVEY,  RESULTING 
FROM  OR  CONNECTED  WITH  THE  TELEGRAPHIC  LONGI- 
TUDE WORK  OF  THE  SURVEY. f 

By  direct  comparison  at  Washington,  D.  C.,  1890,  Sept. 

17,  18,  19 S  —  P  =  +  os.266 

By  direct  comparison  at  St.  Louis,  Mo.,  1890,  Nov.  4,  15  -f-  o  .278 

From  interchange  of  observers  during  longitude  determinations,  after'  one- 
half  of  the  work  was  completed,  generally  from  4  or  5  days'  results: 
Cape  May,   N.  J.,  and  Albany,  N.   Y.,  1891, 

May  and  June S  —  P  = -\-  OM84  ±  o".oii 

Detroit,  Mich.,  and  Albany,  N.  Y.,  1891,  June 

and  July -+-  o  .140  ±  o  .008 

Chicago,  111.,  and  Detroit,  Mich.,  1891,  July.  -f-  o  .172  ±  o  .006 

Minneapolis,  Minn.,  and  Chicago,  111.,  1891, 

Aug +o  .161  ±  o  .010 

Omaha,  Neb.,  and  Minneapolis,  Minn.,  1891, 

Aug.  and  Sept -f-  o  .176  ±  o  .on 

Los  Angeles, Cal.,  and  San  Diego,  Cal.,  1892, 

Feb.  and  Mar -j-  o  .160  ±  o  .006 

*  See  Coast  and  Geodetic  Survey  Report,  1880,  p.  241. 
f  For  these  data  the  author  is  indebted  to  the  Superintendent  of  the 
Coast  and  Geodetic  Survey. 


250  GEODETIC  ASTRONOMY.  §  243. 

San  Diego, Cal.,  and  Yuma,  Ariz.,  1892,  March  -|-  o  .192  ±  o  .004 

Los  Angeles,  Cal.,  and  Yuma,  Ariz.,  1892, 

Mar.  and  April -(-  o  .140  ±  o  .002 

Yuma,  Ariz.,  and  Nogales,  Ariz.,  1892,  April  -j-  o  .150  ±  o  .005 

Nogales,  Ariz.,  and  El  Paso,  Tex.,  1892, 

April  and  May -|-  o  .126  ±  o  .004 

Helena,  Mont.,  and  Yellow  Stone  Lake, 

Wyo.,  1892,  June  and  July -f-  o  .109  ±  o  .010 

El  Paso,  Tex.,  and  Little  Rock,  Ark.,  1893, 

Feb.  and  March -j-  o  .082  ±  o  .010 


The  following  values  depend  on  unrevised  field  com- 
putation : 

Key   West,   Fla.,  and    Charleston,    S.   C.,    1896, 

Feb.  and  March S  —  P  =  +  o'.i47 

Atlanta,  Ga.,  and  Key  West,  Fla.,  1896,  March.  .  -f  o  .121 

Little  Rock,  Ark.,  and  Atlanta,  Ga.,  1896,  April.  -f  o  .130 

Charleston,  S.  C..  and  Washington,  D.  C.,  1896, 

April  and  May -)-  o  .183 

Washington,  D.  C.,  and  Cambridge,  Mass.,  1896, 

May  and  June -f  o  .  142  ±  o".oi3 

Washington,,     D.     C.,     Naval    Observatory  and 

Washington,  D.  C.,  Coast  and  Geodetic  Survey 

Office,  1896,  June  and  July -f-  o  .117  ±  o  .008 

Note  that  the  period  covered  by  this  record  is  nearly  six 
years,  and  that  the  localities  show  that  the  observers  were 
surely  submitted  to  a  great  variety  of  climatic  conditions. 
Yet  if  the  first  two  determinations,  made  when  Mr.  Putnam 
was  comparatively  new  to  the  work,  be  omitted,  the  total 
range  of  the  results  is  only  o8.  no,  It  must  be  remembered, 
however,  that  each  of  these  results,  except  the  first  two, 
depend  upon  from  eight  to  ten  nights  of  observation,  four  or 
five  nights  each  before  and  after  the  interchange  of  observers. 
It  is  quite  probable  that  the  actual  variation  of  the  relative 
personal  equation  from  night  to  night  is  somewhat  greater 
than  that  shown  above. 


§  245-  DISCUSSION  OF  ERRORS.  25 1 

244.  The  absolute  personal  equation  is  the  time  interval 
required  for  the  nerves  and  portions  of  the  brain  concerned  in 
an  observation  to  perform  their  offices.     Although  the  per- 
sonal equation  has  been  studied  by  many,  little  more  can  be 
confidently  said  in  regard  to  the  laws  which  govern  its  magni- 
tude than  that  it  is  a  function  of  the  observer's  personality, 
that  it  tends  to  become  constant  with  experience,  and  that 
probably  whatever  affects  the  observer's  physical  and  mental 
condition  affects  its  value.      But  so  little  is  known  in  regard 
to  it,  that  no  observer  will  predict,  before  the  observations  of 
a  night  have  been  computed,  that  his  personal  equation  was 
large  or  small  on  that  particular  night. 

Discussion  of  Errors. 

245.  Returning  to  a  consideration    of  the   errors   of  the 
telegraphic  longitudes,  it  may  be  said  that  the  ten  results  for 
a  station,  after  eliminating  the  personal  equation,  still  show  a 
range,  ordinarily,  in  primary  work,  of  from  o8. 10  or  less,  to 
os.2O.     This  range  is  larger  than  is  to  be  accounted  for  by  the 
accidental  errors  of  observation,  or  by  any  of  the  other  errors 
enumerated  in  §  242,  except  perhaps  those  of  the  second  and 
third  classes.     Those  most  familiar  with  the  observations  are 
apt  to  account  for  the  large  range  as  arising  either  from  varia- 
tion in  the  personal  equation  or  from  lateral  refraction.     One 
observer  of  long  experience  is  inclined  to  suspect  the  striding 
level  of  giving  errors  which  tend  to  be  constant  for  the  night. 
To  whatever  these  errors  may  be  due,  they  seem  to  be  fairly 
well  eliminated  from  the  mean  for  the  station.      For  in  the 
great  network  of  longitude  determinations,  made  by  the  Coast 
and  Geodetic  Survey,  covering  the  whole  United  States,  the 
discrepancies  arising  in  closing  the  various  "  longitude  tri- 
angles "  are  always  less  than  OMO.* 

*  For  a  good  example  of  a  check  showing  the  degree  of  accuracy  of  this 
network  see  Coast  and  Geodetic  Survey  Report,  1894,  p.  85. 


GEODETIC  ASTRONOMY.  §  247. 


Personal  Equation. 

246.  If,    in   making  a   longitude    determination  circum- 
stances prevent  the  interchange  of  observers,  the  effect  of  the 
relative  personal  equation  upon  the  computed  longitude  may 
still  be  eliminated,  in  part  at  least,  by  a  special  determination 
of  the  equation  by  joint  observations  at  a  common  station. 
The  two  observers  may  place  their  instruments  side  by  side 
in  the  same  observatory,  observe  the  same  stars,  and  record 
their  observations  upon  the  same  chronograph.     The  differ- 
ence of  the  two  chronometer  corrections  computed  by  them, 
corrected  for  the  minute  longitude  difference  corresponding 
to  the  measured  distance  between  their  instruments,  is  then 
their  relative  personal  equation.     Or,  they  may  observe  with 
the  same  transit  as  follows:  On  the  first  star  A  observes  the 
transits  over  the  lines  of  the  first  half  of  the  reticle,  and  then 
quickly  gives  place  to  B,  who  observes  the  transits  across  the 
remaining  lines.     On  the  second  star  B  observes  on  the  first 
half  of  the  reticle,  and  A  follows.     After  observing  a  series 
of  stars  thus,  each  leading  alternately,   each  observer  com- 
putes for  each  star,  from  the  known  equatorial  intervals  of  the 
lines  and  from  his  own  observations,  the  time  of  transit  of 
the  star  across   the   mean   line   of   the   whole  reticle.     The 
difference  of  the  two  deduced  times  of  transit  across  the  mean 
line  is  the  relative  personal  equation.      If  each  has  led  the 
same  number  of  times  in  observing,  the  mean  result  is  inde- 
pendent of  any  error  in  the  assumed  equatorial  intervals  of 
the  lines.     No  readings  of  the  striding  level  need  be  taken, 
and  the  result  is  less  affected  by  the  instability  of  the  instru- 
ment than  in  the  other  method. 

247.  In  certain  cases  in  which  it  is  not  feasible  to  use  a 
telegraph  line  for  a  longitude  determination,  the  same  prin- 
ciples may  be  used  with  the  substitution  of  a  flash  of  light 


§  248.  LONGITUDE  BY   CHRONOMETERS.  2$$ 

between  stations  in  the  place  of  the  electric  wave.  For 
example,  one  might  so  determine  the  longitudes  of  the 
Aleutian  Islands  of  Alaska,  the  successive  islands  being  in 
general  intervisible. 

Longitude  by  Chronometers — Equipment. 

248.  If  a  telegraph  line  is  not  available  between  the  two 
stations,  the  next  method  in  order  of  accuracy,  aside  from 
the  flash  method  alluded  to  above,  is  that  of  transporting 
chronometers  back  and  forth  between  them.  The  transported 
chronometers  then  perform  the  same  duty  as  the  telegraph, 
namely,  that  of  comparing  the  local  times  of  the  two  stations. 

The  chronometric  method  may  perhaps  be  best  explained 
by  giving  a  concrete  example.  The  longitude  of  a  station  at 
Anchorage  Point,  Chilkat  Inlet,  Alaska,  was  determined  in 
1894,  by  transportation  of  chronometers  between  that  station 
and  Sitka,  Alaska,  of  which  the  longitude  was  known.  At 
Anchorage  Point  observations  were  taken  on  every  possible 
night  from  May  I5th  to  August  I2th,  namely,  in  53  nights, 
by  the  eye  and  ear  method,  with  a  transit  of  the  type  shown 
in  Fig.  n,  using  as  a  hack  for  the  observations  chronometer 
Bond  380  (sidereal).  At  the  station  there  were  also  four 
other  chronometers,  two  sidereal  and  two  mean.  These  four 
were  never  removed  during  the  season  from  the  padded 
double- walled  box  in  which  they  were  kept  for  protection 
against  sudden  changes  of  temperature,  and  in  which  the  hack 
chronometer  was  also  kept  when  not  in  use.  The  instru- 
mental equipment  at  Sitka  was  similar.  A  sidereal  chro- 
nometer was  used  as  an  observing  hack,  and  two  other 
chronometers,  one  sidereal  and  one  mean,  were  used  in 
addition.  Nine  chronometers,  eight  keeping  mean  time  and 
one  sidereal  time,  were  carried  back  and  forth  between  the 
stations  on  the  steamer  Hassler. 


254  GEODETIC  ASTRONOMY.  §  249. 

Longitude  by  Chronometers — Observations. 

249.  Aside  from  the  time  observations  the  procedure  was 
as  follows:  Just  before  beginning  the  time  observations  at 
Anchorage  Point  and  again  as  soon  as  they  were  finished,  on 
each  night,  the  hack  chronometer  No.  380  (sidereal)  was 
compared  with  the  two  mean  time  chronometers  by  the 
method  of  coincidence  of  beats,  to  be  described  later  (§250). 
These  two  were  then  each  compared  with  each  of  the  two 
remaining  (sidereal)  chronometers  at  the  station.  These  com- 
parisons, together  with  the  transit  observations,  served  to 
determine  the  error  of  each  chronometer  on  local  time  at  the 
epoch  of  the  transit  observations.*  Whenever  the  steamer 
first  arrived  at  the  station,  and  again  when  it  was  about  to 
leave,  the  hack  chronometer  No.  380  was  compared  with  the 
other  station  chronometers  as  indicated  above,  was  carried  on 
board  the  steamer  and  compared  with  the  nine  steamer 
chronometers,  and  then  immediately  returned  to  the  station 
and  again  compared  with  the  four  stationary  station  chro- 
nometers. As  an  extra  precaution  both  the  station  observer 
and  the  observer  in  charge  of  the  steamer  chronometer  made 
each  of  these  comparisons.  In  the  comparisons  on  the 
steamer,  the  hack  (380)  was  compared  by  coincidence  of  beats 
with  each  of  the  eight  mean  time  chronometers,  and  the 
remaining  (sidereal)  chronometer  was  then  compared  with 
some  of  the  eight.  The  comparisons  on  shore  before  and 
after  the  trip  to  the  steamer  served  to  determine  the  error  of 
the  hack  (380)  at  the  epoch  of  the  steamer  comparisons. 
The  steamer  comparisons  determined  the  errors  of  each  of  the 

*  The  station  chronometers  were  also  intercompared  on  days  when  no 
observations  were  made.  But  this  was  merely  done  to  ascertain  their  per. 
formance,  and  these  comparisons  were  not  used  in  computing  the  longi- 
tude. 


§250.  LONGITUDE  BY   CHRONOMETERS.  255 

steamer  chronometers  on  Anchorage  Point  time.  Similar 
observations  were  made  at  Sitka  to  determine  the  errors  of 
the  nine  steamer  chronometers  on  Sitka  time  as  soon  as  they 
arrived,  and  again  just  before  they  departed  from  Sitka. 
During  the  season  the  steamer,  which  was  also  on  other  duty, 
made  seven  and  a  half  round  trips  between  the  stations.  The 
distance  travelled  was  about  400  statute  miles  for  each  round 
trip. 

250.  The  process  of  comparing  a  sidereal  and  a  mean 
time  chronometer  is  analogous  to  that  of  reading  a  vernier. 
The  sidereal  chronometer  gains  gradually  on  the  mean  time 
chronometer,  and  once  in  about  three  minutes  the  two  chro- 
nometers tick  exactly  together  (one  beat  =  os.5).  Just  as  one 
looks  along  a  vernier  to  find  a  coincidence,  so  here  one  listens 
to  this  audible  vernier  and  waits  for  a  coincidence.  As  in 
reading  a  vernier  one  should  also  look  at  lines  on  each  side 
of  the  supposed  coincidence  to  check,  and  perhaps  correct, 
the  reading  by  observing  the  symmetry  of  adjacent  lines,  so 
here  one  listens  for  an  approaching  coincidence,  hears  the 
ticks  nearly  together,  apparently  hears  them  exactly  together 
for  a  few  seconds,  and  then  hears  them  begin  to  separate,  and 
notes  the  real  coincidence  as  being  at  the  instant  of  symmetry. 
The  time  of  the  coincidence  is  noted  by  the  face  of  one  of  the 
chronometers.  Just  before  or  just  after  the  observation  of 
the  coincidence  the  difference  of  the  seconds  readings  of  the 
two  chronometers  is  noted  to  the  nearest  half-second  (either 
mentally  or  on  paper).  This  difference  serves  to  give  the 
seconds  "reading  of  the  second  chronometer.  The  hours  and 
minutes  are  observed  directly.  When  a  number  of  chronom- 
eters are  to  be  intercompared,  the  experienced  observer  is 
able  to  pick  out  from  among  them  two  that  are  about  to 
coincide ;  he  compares  those ;  selects  two  more  that  are  about 
to  coincide  and  compares  them,  and  so  on;  and  thus  to  a 


256  GEODETIC  ASTRONOMY.  §  252. 

certain  extent  avoids  the  waits — of  a  minute  and  a  half  on 
an  average — which  would  otherwise  be  necessary  to  secure 
an  observation  on  a  pair  of  chronometers  selected  arbitrarily. 

Computation  of  a  Longitude  by  Chronometers. 

251.  The  following  example  (taken  from  another  set  of 
observations)  will  show  how  the  chronometer  comparisons  are 
computed.  A  certain  Dent  mean  time  chronometer  was 
compared  with  a  certain  Negus  sidereal  chronometer  on  Oct. 
14,  1892,  at  a  station  2h  I2m  west  of  Washington.  It  was 
found  that  uh  2Om  23s. o  A.M.  Dent  =  I2h  54™  41*. o  Negus. 
The  correction  to  the  Dent  on  local  mean  time  was  known 
to  be  —  2h  nm  53s. 41,  and  the  correction  of  the  Negus  to 
local  sidereal  time  was  required. 

Time  by  Dent  chronometer 23*  2Om  23". oo 

Correction  to  Dent —  2     n     53.41 


Local  mean  solar  time 21     08     29.59 

Reduction  to  sidereal  interval  (§  290) +03     28  .38 


Sidereal  interval  from  preceding  mean  noon 21     n     57.97 

"        time  of  preceding  mean  noon  (Oct.  13). .        13     31     14  .05 


Local  sidereal  time 10     43     12  .02 

Time  by  Negus  chronometer 12     54     41.00 


Correction  to  Negus  chronometer „ —2     n     28.98 

The  computation  is  modified  in  an  obvious  manner  if  it  is 
the  error  of  the  sidereal  chronometer  that  is  known. 

252.  This  process  of  comparing  chronometers  is  so  accu- 
rate, that  it  was  found  that  the  two  values  of  the  error  of 
either  of  the  station  sidereal  chronometers,  as  derived  from 
the  comparisons  described  above  in  two  different  ways  from 
the  hack  chronometer,  seldom  differed  by  more  than  os.O3. 
This  corresponds  to  an  error  of  n8  in  noting  the  time  of 


§  254-  LONGITUDE   BY   CHRONOMETERS. 

coincidence  of  beats,  on  the  supposition  that  all  the  error  was 
made  in  one  of  the  four  comparisons  concerned.  If  two 
chronometers  of  the  same  kind,  both  sidereal  or  both  mean 
time,  are  compared  directly  it  requires  very  careful  observing 
to  secure  their  difference  within  os.  10  of  the  truth. 

253.  The  comparisons  of  the  other  four  station  chrono- 
meters with  the  hack  chronometer  immediately  before  and 
after  transit  observations  gave  the  errors  of  each  of  those  four 
chronometers.     To  compute  the  errors  of  the  steamer  chro- 
nometers at  the  time  of  the  comparisons  made  on  the  steamer, 
it  is  first  necessary  to  secure  as  good  a  determination  as  possi- 
ble of  the  error  of  the  hack  chronometer  at  the  epoch  of  those 
comparisons.      One  value  for  that  error  was  obtained  in  an 
obvious  manner  by  assuming  that  the  hack  chronometer  ran 
at  a  uniform  rate  between  the  last  preceding  and  the  next 
following  transit  observations.      Four  other  determinations  of 
the  error  of  the  hack  at  that  epoch  were  obtained,  by  making 
that   same   assumption   for    each    of    the    other  four  station 
chronometers,  and   deriving  the   error  of  the  hack  from   the 
comparisons  made  with  that  chronometer  at  the  station  before 
and  after  the  steamer  comparisons.      The  weighted  mean  of 
these  five  values  of  the  error  of  the  hack  was  used.      For  the 
method  of  deriving  the  relative  weights  which  were  assigned 
to   these   five    results   see   §  260.      At   Anchorage    Point    13 
comparisons  were  made  with  the  steamer  chronometers.      In 
six  cases  out  of  the  thirteen  the  range  of  the  five   derived 
values  of  the  error  of  the  hack  was  less  than  os.2. 

254.  Having  now  the  errors  of  the  steamer  chronometers 
on  the  local  time  of  each  station  at  the  time  of  arrival  at  and 
departure  from  each  station,  the  difference  of  longitude  was 
computed  in  the  manner  indicated  by  the  following  illustra- 
tion.    Suppose  chronometer  No.  231  to  have  been  found  to 
have  the  following  errors  on  a  certain  round  trip: 


258  GEODETIC  ASTRONOMY.  §  256. 

Anchorage  Point,  at  departure,  May  15,  Qh  oom  A.M.  40"  fast  of  A.  P.  time. 

Sitka,  on  arrival May  16,  9  oo  A.M.  n     "      "  Sitka     " 

Sitka,  at  departure May  22,  9  oo  A.M.    4     "      "Sitka     " 

Anchorage  Point May  23,  9  oo  A.M.  32     "      "  A.  P.     " 

From  the  two  Anchorage  Point  observations  it  appears 
that  the  chronometer  has  lost  88  in  the  eight  days  it  was  gone 
from  there.  From  the  two  Sitka  observations  it  appears  that 
7s  were  lost  while  at  Sitka.  Hence  the  chronometer  lost  Is 
only  while  travelling  both  ways  between  the  stations,  or  its 
travelling  rate  was  O8.5  per  day,  losing.  Applying  this  rate 
to  the  errors  as  determined  at  Anchorage  Point,  we  find  that 
the  errors  of  the  chronometer  on  Anchorage  Point  time  at  the 
epochs  of  the  Sitka  steamer  comparisons  were  39s.  5  fast  and 
32s. 5  fast,  and  that  the  difference  of  longitude  required  is 
398.5  —  ns.o  =  32s. 5  —  4s. o  =  28*. 5,  A.  P.  west  of  S.  We 
have  thus  derived  the  longitude  difference  on  the  supposition 
that  the  steamer  chronometers  have  a  travelling  rate  which  is 
constant  during  the  round  trip,  and  without  any  assumptions 
as  to  the  rates  while  in  port.  The  assumptions  as  to  the 
station  chronometers  have  been  simply  that  each  preserves  a 
constant  rate  between  successive  transit  observations. 

255.  The  longitude  was  thus  computed  from  each  round 
trip  starting  from  Anchorage  Point,  and  the  mean  taken.      If 
the  chronometers   had    continually  accelerated  (or  retarded) 
rates,  this  mean  was  subject  to  an  error  arising  from  that  fact. 
To  eliminate  such  a  possible  error,  and  to  serve  as  a  check 
upon  the  computation,  a  second  computation  was  made  from 
each  round  trip  starting  from  Sitka,   and  the  mean  taken. 
The   error   from   acceleration   (or  retardation)    of    rates    was 
necessarily  of  opposite  sign  in  this  mean.     The  mean  of  these 
two  results  is  then  subject  only  to  accidental  errors,  in  so  far 
as  the  chronometers  are  concerned. 

256.  The    following    table    shows    the    separate    results 
obtained  and  the  manner  of  combining  them : 


§257- 


LONGITUDE  BY   CHRONOMETERS. 


259 


DIFFERENCE  OF    LONGITUDE,  IN   SECONDS,  BETWEEN   SITKA 
AND    ANCHORAGE    POINT,  CHILKAT    INLET,  ALASKA. 

SUMMARY    OF    RESULTS    FROM    SEVEN    ROUND    TRIPS,    STARTING    FROM 
ANCHORAGE    POINT,    CHILKAT    INLET. 


Chronometers, 
M.  T.  or  Sid. 

i't           2d            3d          4th          5th           6th           ?th 

Means. 
AA 

Weights. 

M.  T.             231 

28.03      26.36      28.36      28.19      28.45      28.19      28.18 

27.97 

3 

1507- 

28.44      29.06      29.18      28  26      28.27      28.20      28.54 

28.56 

4 

1510 

28.57      29.25      29.00      28.52      28.63      28.06      28.58 

28.66 

7 

196 

28.59      29.09      29.54      28.59      28-43      28.51      28.92 

28.81 

3 

1542 

28.11      28.11      28.66      28.23      28.47      28.38      28.37 

28.33 

22 

1728 

28.66      28.94      29.16      28.63      28.58      28.43      28.59 

28.71 

6 

208 

27.95      27.40      28.21      28.19      28.42      28.42      28.09 

28.10 

6 

2T67 

28    21           28.56           28.90          28.55           28.68          28.27          28.64 

28.54 

*7 

Sid.                387 

28.20          28.44          28.91           27.93           28.41           27.93          28.59 

28.34 

6 

Mean 
Weighted  mean 

28.31           28.36          28.88           28.34          28.48          28.27          28.50 

28.45 
28.44 

Weight. 

3I222I2 

Weighted  mean    oh    cm    28*. 44  ±  o".o5 
SUMMARY    OF    RESULTS    FROM    SEVEN    ROUND    TRIPS,  STARTING    FROM    SITKA. 


Chronometers, 
M.  T.  or  Sid. 

,«           2d            3d           4th           5tn           6.h           ?th 

Means. 
AA 

Weights. 

M.  T.             231 

28.87       28.78       28.74       28.39       28.37       28.71       28.11 

28.57 

3 

1507 

27.69       29.08       29.11       27.76       28.78       27.93       28.64 

28.43 

4 

1510 

28.37       28.88       28.82       27.91       28.83       28.10       28.58 

28.^0 

7 

196 

28.59       29.07       28.95       27.66       28.03       29.56         9.20 

28.72 

3 

1542 

28.93       28.57       28.59       28.22       28.50       28.50         8.32 

28.52 

22 

1728 

27.59       28.90       28.75       27.99       29.01       28.09         8.75 

28.44 

6 

208 

27.71       28.03       28.52       28.58       27.88       28.76         7.65 

28.16 

6 

2167 

28  24       28.71       28.80       28.27       28.77       28.31         8.49 

28.51 

17 

Sid.                387 

28.68       28.80       28.43       27.69       28.97       27.98         8.73 

28.47 

6 

Mean 
Weighted  mean 

28.30       28.76       28.75       28.05       28.57       28.44       28.50 
28.41       28.69       28.70       28.13       28.61       28.38       28.44 

28.48 
28.48 

Weight 

1222222 

Weighted  mean    oh    om    28».48  ±  os.os 

Final  mean  AA  =  -f-  oh  oom  28". 46  .±  o8.os 

Longitude  of  Sitka,  9    01    21  .48  ±  o  .13 

Longitude  of  Anchorage  Point  9    01    49  .94  ±  o  .14 

or          135°  27'  29". 10  ±  2".io 

257.  The  steamer  started  from  Anchorage  Point  at  the 
beginning  of  the  season,  and  finished  at  Sitka  at  the  season's 
end,  after  7^  round  trips.  The  last  half-trip  was  omitted  in 


260  GEODETIC  ASTRONOMY.  §  259. 

the  first  part  of  the  above  computation,  and  the  first  half-trip 
omitted  in  the  second  part.  If  there  had  been  simply  seven 
round  trips  starting  from  Anchorage  Point  the  procedure 
would  have  been  to  deal  regularly  with  all  trips  in  the  first 
half  of  the  computation;  and  in  the  last  half  in  addition  to 
the  six  regular  round  trips  starting  from  Sitka,  the  last  half- 
trip  (S.  to  A.  P.)  and  the  first  half-trip  (A.  P.  to  S.)  would 
have  been  used  together  as  a  seventh  round  trip  from  Sitka. 

258.  Let   N  be   the   number  of  days  during  which   the 
chronometers  were  depended  upon  to  carry  the  time  during 
each  round  trip,  reckoned  as  follows:  Add  together  the  two 
intervals  between  comparisons  of  the  steamer  chronometers 
with  the  shore  chronometer  at  the  beginning  and  at  the  end 
of   each   half-trip,   and  increase  this  by  adding  the  interval 
from    each    comparison    of   the   observing   chronometer    and 
steamer  chronometers  to  the  nearest  transit  observations  made 
at   that   station.     The   weight  assigned  to  each  trip   in   the 
above  computation  is  proportional  to  i/N. 

259.  What  relative  weights  shall  be  assigned  to  the  results 
from   the   different   chronometers  ?     Some  evidently   run    at 
a  more  nearly  constant  rate  than  others.     Let  /,,  /2,  /3,  .  .  .  4 
be  the  separate  values  of  the  longitude  as  given  by  any  one 
chronometer,  and  lm  their  mean,  and  let  n  be  the  number  of 
such  values,  or  the  number  of  trips.     Then  by  least  squares 
the  probable  error  of  any  one  value  is 

'.  -  4J'  +  i/a  -  LY  .  •  •  (4r=="£)r] 

n  —  i 

By  the  rule  that  the  weight  of  a  result  is  inversely  propor- 
tional to  the  square  of  its  probable  error,  the  relative  weights 
to  be  assigned  to  the  chronometers  are  proportional  to 

n  —  i 

v  -      •     (I21) 


[(A  -  /«)'  +  tt  -  «T ...  (4  - 


§  26 1.  LONGITUDE  BY   CHRONOMETERS.  26 1 

The  factor  0.455  is  dropped  for  simplicity  since  we  are  deal- 
ing with  relative  weights  only.  In  the  above  computation 
the  sum  [(/,  -  l^  +  (/,-/„)*  ...(/,-  /w)a]  was  determined 
from  each  half  of  the  computation,  and  the  mean  used  in  the 
denominator  of  (121).  The  remainder  of  the  computation 
needs  no  explanation. 

260.  The  relative  weights  assigned  to  the  station  chro- 
nometers as  indicated  in  §  253  may  be  determined  by  an 
analogous  process.  Let  o  be  the  error  of  a  chronometer  at 
the  epoch  of  the  transit  time  observations  as  determined  from 
those  observations.  Let  /  be  its  error  at  that  same  instant 
interpolated  between  its  errors  as  determined  at  the  last  pre- 
ceding and  first  following  transit  time  observations  on  the 
assumption  that  its  rate  during  that  interval  is  constant. 
Then  /—  o  is  a  measure  of  the  behavior  of  the  chronometer. 
It  is  the  amount  by  which  the  chronometer  has  gone  wrong 
on  the  supposition  that  the  transit  observations  may  be  con- 
sidered exact.  The  chronometer  apparently  indicates  that 
the  station  at  the  middle  observation  was  at  a  distance  /  —  o 
in  longitude  from  its  position  at  the  preceding  and  following 
observations.  For  a  group  of  chronometers  whose  errors  are 
all  determined  a  number  of  times  in  succession  by  the  same 
transit  observations,  the  relative  weights  are  evidently  pro- 
portional to  the  quantities 


261.  The  above  example  serves  to  illustrate  the  principles 
involved  in  the  computation  of  a  longitude  by  chronometers. 
The  accuracy  of  the  derived  longitude  is  greater,  the  greater 
the  number  of  chronometers  used,  the  greater  the  number  of 
trips,  the  smaller  the  average  value  of  N  (§  258),  and  of 
course  depends  intimately  upon  the  quality  of  the  chronom- 


262  GEODETIC  ASTRONOMY.  §  263. 

eters  and  the  care  with  which  they  are  protected  from  jars 
and  from  sudden  changes  of  temperature.  Unless  the  round 
trips  are  quite  short  the  errors  of  the  transit  time  observations 
will  be  small  as  compared  with  the  other  errors  of  the  process. 
If  considered  necessary  the  relative  personal  equation  of  the 
observers  may  be  eliminated  from  the  result  by  the  same 
methods  that  are  used  in  connection  with  telegraphic 
determinations  of  longitude. 

262.  If  the  trips  are  very  long,  it  may  possibly  be  advis- 
able to  determine,   by  a  special  series  of  observations,   the 
temperature    coefficient    of    each    chronometer    and    also    a 
coefficient  expressing  its  acceleration  (or  retardation)  of  rate, 
and   to    apply   corresponding    computed    corrections   to    the 
travelling   rates.*     The   chronometers   are   compensated    for 
temperature  as  far  as  possible  by  the  maker,  of  course,  but 
such  compensation  cannot  be  perfect.     The  thickening  of  the 
oil  in  the  bearings  tends  to  increase  the  friction  with  lapse  of 
time,  and  by  diminishing  the  arc  of  vibration  of  the  balance- 
wheel  to  increase  the  rate  of  running.     Attempts  to  use  rate 
corrections  depending  upon  the  computed   coefficients  of  a 
chronometer   have    usually   been    rather   unsatisfactory,    and 
should  not  be  made  except  in  extreme  cases. 

Longitude  Determined  by  Observing  the  Moon. 

263.  If  none  of  the  preceding  methods  are  available,  one  is 
forced  to  use  those  methods  which  depend  upon  the  motion 
of  the  Moon,  or  perhaps  to  observe  upon  Jupiter's  satellites. 

The  place  of  the  Moon  has  been  observed  many  times  at 
the  fixed  observatories.  From  these  observations  its  orbit 
and  the  various  perturbations  to  which  it  is  subject  have  been 
computed.  In  the  American  Ephemeris  and  similar  publica- 

*  For  details  of  this  process  see  Doolittle's  Practical  Astronomy,  pp. 

383-388. 


§265.  LONGITUDE  BY   THE  MOON.  263 

tions,  tables  will  be  found  giving  the  Moon's  right  ascension 
and  declination  for  every  hour,  and  also  other  tables  giving 
its  place  as  defined  in  other  ways.  Suppose  now  that  an 
observer  at  a  station  of  which  the  longitude  is  required 
determines  the  position  of  the  Moon  and  notes  the  local  time 
at  which  his  observation  was  made.  He  may  then  consult 
the  Ephemeris  and  find  at  what  instant  of  Greenwich  time  the 
Moon  was  actually  in  the  position  in  which  he  observed  it. 
The  difference  between  this  time  and  the  local  time  of  his 
observation  is  his  longitude  reckoned  from  Greenwich. 

Among  the  processes  by  which  the  position  of  the  Moon 
may  be  determined  for  this  purpose  are  the  following. 

264.  The  local  sidereal  time  of  transit  of  the  Moon  acrbss 
the  meridian  of  the  station  may  be  observed  with  a  transit, 
and  a  chronometer  of  which  the  error  is  determined  in  the 
usual  way  by  observations  upon  the  stars.      Or,  what  is  in 
principle  the  same  thing,  the  right  ascension  of  the  Moon  may 
be  derived  by  comparing  its  time  of  transit  with  that  of  four 
stars  of  about  the  same  declination  as  the  Moon,  two  transit- 
ing shortly  before  it,  and  two  soon  after  it.      In  either  case 
the  right  ascension  of  the  Moon  at  the  instant  of  its  transit 
may  be  computed,  and  from  the  Ephemeris  the  Greenwich 
time  at  which  the   Moon  had  that  right  ascension  becomes 
known. 

265.  The  lunar  distance  of  a  heavenly  body  is  the  angle 
between  two  lines  drawn  from  the  center  of  the  Earth — one 
to  the  center  of  the  Moon  and  the  other  to  the  center  of  the 
body  considered.      Or,  in  other  words,  it  is  the  angle  between 
the  two  objects  as  seen  from  the  Earth's  center.     The  Ephem- 
eris gives    the  lunar  distances  of    the  Sun,   the    four  larger 
planets,    and   of  certain  stars,    at  intervals    of    three   hours, 
Greenwich  mean  time.     An  observer  anywhere  may  measure 
the   angular  distance   from   the  Moon  to  any  one  of    these 


264  GEODETIC  ASTRONOMY.  §  268. 

objects,  with  a  sextant  or  other  suitable  instrument.  His 
measurement  reduced  to  the  Earth1  s  center  gives  the  lunar 
distance;  from  which  with  the  use  of  the  Ephemeris  the 
Greenwich  time  of  the  observation  becomes  known ;  and  also 
his  longitude  if  he  noted  the  local  time  of  the  observation. 

266.  A  star  is  said  to  be  occulted  during  the  time  it  is  out 
of  sight  behind  the  Moon.     The  beginning  and  end  of  the 
occupation,   that   is,   the  instants   of   disappearance   and    re- 
appearance of  the  star,  called  its  immersion  and  emersion,  are 
phenomena    capable    of    being    observed    with    considerable 
accuracy.      The  Ephemeris  gives  the  necessary  elements  for 
computing   the    Washington   times  of   occultation   of  various 
stars  as  seen  from  any  point  upon  the  surface  of  the  Earth. 
The  local  time  of  the  occultation  being  observed,  either  of  the 
immersion  or  emersion,  and  the  Washington  time  being  com- 
puted, the  longitude  becomes  known.     An  observation  of  the 
local  times  of  the   phenomena  of  an  eclipse  of  the  Sun   or 
Moon  furnishes  a  similar  determination  of  longitude. 

267.  The  computations  required  in  the  last  two  methods 
are   quite  long  and   complicated,   and   the  theories  involved 
require  much  study  for  their  mastery.      The  method  of  cul- 
minations  gives   rise   also   to   rather    difficult   computations, 
though  not  so  difficult  as  those  just  mentioned.      However, 
the   time   and   labor   expended   would   be   fully   rewarded   if 
accurate  results  were  obtained.      But  any  of  these  methods 
give  rise  to  results  which  are  crude  in  comparison  with  those 
given   by   the   telegraphic   method,    or  by  transportation   of 
chronometers.       The    method    of    occultations    requires    the 
greatest  amount   of  computing,   but  also  gives   the  greatest 
accuracy,  of  the  methods  named. 

268.  Three  conditions  stand  in  the  way  of  the  attainment 
of  accuracy  by  any  method  involving  the  Moon.     Firstly,  the 
Moon  requires  about  2/J  days  to  make  one  complete  circuit 


§  2/O.  LONGITUDE  B  Y  THE  MOON.  265 

in  its  orbit  about  the  Earth.  The  apparent  motion  of  the 
Moon  among  the  stars  is  then  about  one-twenty-seventh  as 
fast  as  the  apparent  motion  of  the  stars  relative  to  an 
observer's  meridian,  which  furnishes  his  measure  of  time. 
Any  error  in  determining  the  position  of  the  Moon  is  then 
multiplied  by  at  least  twenty-seven  when  it  is  converted  into 
time  in  the  progress  of  the  computation.  If  then  the  time  of 
transit  of  the  Moon,  for  example,  could  be  observed  as 
accurately  as  that  of  a  star,  one  would  expect  the  errors  in  a 
longitude  computed  from  Moon  culminations  to  be  twenty- 
seven  times  as  great  as  the  errors  of  the  local  time  derived 
from  the  same  number  of  star  observations. 

269.  Secondly,  the  motion  of  the  Moon  is  so  difficult  to 
compute    that    its    positions    at   various    times    as    given    in 
the  Ephemeris,  and  also  of   course  the  data  there  given  in 
regard    to    lunar  distances  and  occultations,  are  in  error  by 
amounts  which  become  whole  seconds  when  multiplied  by  the 
factor  twenty-seven.      This  source  of  error  is  often  avoided 
in  the  method  of  Moon's  transits  (culminations)  by  using  in 
the  computation  for  each  night  the  Moon's  right  ascension  as 
corrected  at  Greenwich,  or  some  other  station  of  known  longi- 
tude, by  direct  observation  on  that  same  night. 

Thirdly,  the  limb,  or  edge  of  the  visible  disk  of  the 
Moon,  is  necessarily  the  object  really  observed,  and  this  is  a 
"ragged  edge"  rather  than  a  perfect  arc,  for  purposes  of 
accurate  measurement. 

270.  The    determination    of    the    points    at    which     the 
boundary  between  Alaska  and  British  America  (i4ist  meri- 
dian) crosses  the  Yukon  and  Porcupine  rivers  was  one  of  the 
comparatively  few   instances  in  late  years   in  which   it   was 
necessary  to  resort  to  observations  upon  the  Moon  to  deter- 
mine an  important  longitude.     To  determine  the  longitude 
by  transportation  of  chronometers  would  have  been  exceed- 


266  GEODETIC  ASTRONOMY,  §  271. 

ingly  difficult  and  costly,  for  there  is  more  than  a  thousand 
miles  of  slow  river  navigation  between  the  mouth  of  the 
Yukon  River  and  either  station.  At  a  station  near  the  point 
where  the  Yukon  crosses  the  boundary,  Moon  culminations 
were  observed  on  23  nights.  Four  of  the  results  were 
rejected  as  worthless.  The  other  19  gave  results  ranging 
from  9h  22m  3Os.o  to  48*.  9,  with  a  weighted  mean  of  38S.5. 
Fourteen  of  these  computed  results  depend  upon  the  Moon's 
place  as  corrected  by  corresponding  observations  at  Greenwich 
or  San  Francisco.  At  the  same  station  two  observed  occul- 
tations  gave  for  the  seconds  of  the  longitude  3  5s.  5  and  378.2, 
and  a  solar  eclipse  gave  32s.  2.  At  a  station  near  the  point 
where  the  Porcupine  River  crosses  the  boundary,  13  observed 
Moon  culminations  computed  by  the  use  of  corresponding 
observations  on  the  same  nights  at  San  Francisco,  Washing- 
ton, or  Greenwich,  gave  longitudes  varying  from  9h  23""  45s.  5 
to  63s.  8,  with  a  weighted  mean  of  5  5s.  4.  One  observed 
occupation,  both  immersion  and  emersion,  gave  for  the 
seconds  of  the  longitude  63s.  6.  These  examples*  will  serve 
to  indicate  roughly  the  possibilities  of  the  lunar  methods  of 
determining  longitude.  Experienced  observers  took  the 
observations  at  both  stations.  It  should  be  noted,  however, 
that  in  such  high  latitudes  (the  stations  were  near  the  Arctic 
Circle),  the  trigonometric  conditions  are  unfavorable  to 
accurate  time  determinations,  and  the  climatic  conditions 
were  such  as  to  make  observing  difficult. 

271.   Those    wishing    to    study   these    lunar    methods    of 
determining  the  longitude  are   referred  for  details  to  Doo 
little's    Practical    Astronomy;    to    Chauvenet's    Astronomy, 
vol.  I.  ;  and  in  the  American  Ephemeris  (aside  from  the  tables) 


*  For  a  more  complete  account  of  these  observations  see   Coast   and 
Geodetic  Survey  Report,  1895,  pp.  331-336. 


§  2^2.  LONGITUDE   B  Y  JUPITER.  267 

especially  to  the  pages,  in  the  back  of  the  volume,  headed 
"  Use  of  Tables." 

272.  The  Ephemeris  gives,  for  each  night  of  the  year 
when  Jupiter  is  not  too  near  the  Sun  to  be  observed,  the 
Washington  mean  time  of  the  occultations  and  eclipses  of 
Jupiter's  satellites  by  that  planet,  and  also  the  transit  of  the 
satellites  and  their  shadows  across  the  face  of  the  planet. 
An  eclipse  may  be  observed  at  a  station  of  which  the  longi- 
tude is  required.  By  comparison  of  the  computed  Washing- 
ton mean  time  of  the  eclipse  as  given  in  the  Ephemeris,  and 
the  observed  mean  time,  the  required  longitude  may  be 
derived.  The  times  of  the  other  phenomena  mentioned  are 
given  to  the  nearest  minute  only.  They  may  be  observed 
simultaneously  by  two  observers  using  the  Ephemeris  merely 
to  indicate  when  to  be  on  the  alert.  The  difference  in  the 
local  times  of  observation  of  the  same  phenomena  is  the 
difference  of  longitude  of  the  observers,  the  transit  or  occulta- 
tion  serving  merely  as  a  signal  that  may  be  seen  at  the  same 
instant  by  both.  The  difficulty  of  accurately  observing  these 
phenomena  (including  the  eclipses)  makes  the  derived  longi- 
tudes only  rough  approximations.  The  time  of  a  satellite 
may,  for  example,  be  observed  a  whole  minute  sooner  than  it 
actually  occurs,  if  a  low-power  telescope  is  used.  Such  errors 
may  be  partially  eliminated  by  observing  the  reappearance  as 
well  as  the  disappearance. 


268  GEODETIC  ASTRONOMY.  §  2/3. 


CHAPTER   VIII. 
MISCELLANEOUS. 

Suggestions  about  Observing. 

273.  Among  the  characteristics  of  a  good  observer,  that 
is,  of  an  observer  who  will  secure  the  maximum  accuracy 
with  a  given  expenditure  of  time  and  money,  in  making  such 
astronomical  determinations  as  are  treated  in  this  book,  may 
be  mentioned  the  following: 

He  is  without  bias  as  to  the  results  to  be  obtained,  his 
prime  motive  being  always  to  come  as  near  as  possible  to  the 
truth.  He  has  that  kind  of  self-control  which  makes  it  possi- 
ble for  him  to  prevent  the  knowledge  that  the  result  he  is 
securing  is  too  small  (or  too  large)  to  check  with  other 
determinations,  from  having  the  slightest  effect  upon  his 
observations.  For  example,  he  may  know  that  his  observa- 
tions, in  making  a  telegraphic  determination  of  the  longitude 
of  a  station,  are  placing  that  station  os.5  farther  west  than  it 
has  been  fixed  by  a  primary  triangulation,  and  yet  have  no 
tendency  to  observe  stars  earlier  or  later  than  usual.  Or, 
when  in  reading  a  micrometer  upon  an  azimuth  mark  several 
times  in  quick  succession  he  secures  three  or  four  readings 
which  agree  almost  exactly,  and  then  one  which  differs  from 
them  by  two  seconds  (say),  thus  making  a  bad  looking  break 
in  his  record,  he  will  not  suppress  or  "  spring  "  this  reading, 
though  it  may  serve  to  make  him  more  careful  with  following 
readings. 


§  2/6.  OBSERVING.  269 

274.  He  is  well  aware  of  the  minuteness  of  the  allowable 
errors.     A  student,  when  warned  that  he  must  not  apply  any 
longitudinal  force  to  the  head  of  the  micrometer  of  a  zenith 
telescope,  will  perhaps  experiment  for  himself,  by  purposely 
applying  a  little  pressure  while  making  a  bisection,  and  not 
being  able   to   see  any  appreciable  motion,  will  become  in- 
credulous   as    to    the    necessity    of    the    warning.      A    good 
observer,  on  the  other  hand,  knows  that  he  can  secure  obser- 
vations such  that  the  combination  of  errors  from  all  sources 
produce  an  error  in  each  result  which  is  as  apt  to  be  less  than 
o".3    as    greater    than    that   value    (e  =  ±  (/'.jo),    although 
o".3  is  a  fraction  of  the  apparent  width  of  the  line  with  which 
he  makes  the  bisection.     In  other  words,  he  knows  that  he 
can  make  pointings  under  good  conditions,  of  which  the  errors 
are  so  small  as  to  be  invisible  in  the  telescope.      He  knows 
that  he  can  make  pointings  with  a  probable  error  of  ±  o".5, 
say,  with  a  telescope  with  which  it  would  be  hopeless  to  try 
to  see  a  rod  one-sixth  of  an  inch  in  diameter  placed  one  mile 
away  Q-  inch  subtends  o".$  at  one  mile). 

275.  He  is  conscious  that  the  most  delicate  manipulation 
is  required.      He  knows  that  his  instrument  is  built  of  elastic 
material,  and  that  unless  he  is  exceedingly  careful  to  apply 
only   such   forces  as   are  necessary  he  may  readily   produce 
deformations    in    his    instrument,    which    though    strictly    in 
accordance   with    the   modulus  of   elasticity  of   the   material 
composing  it,  are  yet  as  large  as  the  largest  allowable  errors 
of  observation.      One  may  sometimes  secure  striking  ocular 
evidence  of  this  by  watching  a  bisection,  in  a  reading  micro- 
scope on  a  horizontal  circle  (or  in  the  telescope),  while  a  poor 
observer   makes  his   pointings  with   another   of  the   reading 
microscopes  on  the  instrument. 

276.  A  good  observer  does  not  consider  his  instrument  to 
be   of   fixed   dimensions   or   shape,    even   when   no    external 


2/0  GEODETIC  ASTRONOMY.  §  277. 

forces  are  applied  to  it.  He  knows  that  it  is  constantly 
undergoing  changes  of  shape  due  to  changes  of  temperature ; 
that  these  changes  even  under  the  best  conditions  that  he  can 
secure  may  produce  errors  of  the  same  order  of  magnitude  as 
the  observer's  errors;  and  under  adverse  conditions  may  pro- 
duce errors  which  are  larger  than  all  the  others  concerned  in 
the  measurement. 

With  respect  to  movements  under  stress  and  under 
thermal  changes,  the  support  of  the  instrument  (tripod, 
block,  or  pier)  should  be  considered  as  a  part  of  the  instru- 
ment. 

Suggestions  about  Computing. 

277.  Almost  the  first  question  that  arises  on  commencing 
a  given  kind  of  computation  for  the  first  time  is  "  To  how 
many  decimal  places  must  each  part  of  the  computation  be 
carried  ?"  If  too  few  figures  are  used  the  errors  from  the 
cast  away  decimal  places  become  larger  than  is  allowable.  If 
too  many  places  are  used  the  computation  becomes  slower 
than  is  necessary, — the  work  required  for  interpolations  in 
whatever  tables  are  used  being  especially  liable  to  increase 
rapidly  with  an  increase  of  decimal  places.  A  good  general 
guide  in  this  matter  is  to  carry  each  part  of  every  computation 
to  as  many  decimal  places  as  correspond  to  two  doubtful  fig- 
ures in  the  final  result.  That  is,  when  the  computation  is 
finished,  and  the  probable  error  is  computed,  there  should  in 
general  be  two  significant  figures  in  the  probable  error.  Or, 
in  other  words,  the  probable  error  should  be  between  10  and 
100  units  in  the  last  place.  It  may  be  allowable  to  drop  one 
more  figure  than  above  indicated  if  to  do  so  decreases  the 
work  of  computation  very  much,  as  in  computing  sextant 
observations  for  time  (see  foot-note  to  Example  8,  at  the  end 
of  Chapter  III).  It  is  important,  after  deciding  upon  the 


§  2/9-  COMPUTING.  271 

number  of  places  to  use  in  a  computation,  to  adhere  to  that 
number  strictly.  To  carry  some  numbers  one  place  farther 
than  others,  in  a  column  to  be  added,  is  useless:  and  worse 
than  useless,  for  it  leads  to  mistakes  such  as  adding  tenths 
and  hundredths,  for  instance,  as  if  they  were  in  the  same 
column. 

If  a  number  ends  in  a  five  and  the  last  figure  is  to  be  cast 
away,  shall  the  five  be  called  ten  or  zero  ?  Both  are  equally 
near  the  truth.  A  good  rule  is,  in  such  cases,  to  make  the 
last  retained  figure  even  (not  odd).  This  will  mean  calling 
the  five  a  ten  about  half  of  the  time,  and  avoids  the  constant 
tendency  to  make  the  result  too  large  (or  too  small)  that 
would  exist  if  the  five  were  always  called  ten  (or  zero). 

278.  If  many  astronomical  computations  are  to  be  made, 
Barlow's  tables  of  squares,  etc.,  Crelle's  four-place  multipli- 
cation tables,  and  a  machine  for  multiplying  will  be  found 
convenient  aids, — the  last  two  for  checks,   especially.      For 
example,   apparent   star   places  may  be   computed   by  loga- 
rithms  in   the  usual  way,    and   then  checked   by  a  separate 
computation   by   natural   numbers   and    the    use   of    Crelle's 
tables  (or  a  computing-machine).     The  check  will  be  much 
more  efficient  than  repeating  the  logarithmic  work  because  an 
entirely  different  set  of  figures  are  used,  and  it  will  take  about 
the  same  amount  of  time. 

279.  The  difference  between  a  good  computer  and  a  poor 
one  lies  largely  in  the  industry  and  ingenuity  with  which  a 
good  computer  applies  such  checks  to  his  work  to  find  what- 
ever   mistakes    he    makes.       Rough    checks    should    not    be 
despised,    such   as   comparing   two   computations   which    are 
nearly  alike  (computations  of  two  successive  sets  of  observa- 
tions for  example),  or  such  as  checking  an  exact  computation 
by  formula  (98),  §  193,  by  the  use  of  the  table  in  §  310. 

Means  should  always  be  checked  by  residuals.      If  resid- 


272  GEODETIC  ASTRONOMY.  §  283. 

uals   be  obtained  by  subtracting  a  mean  from   each   of  the 
separate  values,  the  sums  of  the  positive  and  of  the  negative 

residuals  so  obtained  must  not  differ  by  more  than  —  units 

corresponding  to  the  last  place  of  the  mean,  where  n  is  the 
number  of  the  separate  values. 

280.  In   converting  angles  into   time,  or  vice  versa,  it  is 
about  as    rapid    to   use  the    relations   360°  =  24'',    15°  =  ih, 
i°  =  4m,  i'  =  4s,  15"  =  Is,  as  it  is  to  use  the  tables  given  for 
that  purpose  on  page  560  of.  Vega's  Logarithmic  Tables,  and 
elsewhere.      The  tables  may  be  used  as  a  check. 

281.  When  several  computations  of  the  same  kind  are  to 
be  made,  it   usually  saves  time  to  carry  along  corresponding 
portions    together.      For    example,    in    computing    apparent 
places  all  the   star  numbers  may  be  taken  out  at  one  time, 
later  all  the  values  of  log  cos  (G  -f-  «0),  and  so  on. 

The  use  of  a  fixed  form  for  a  computation  saves  time  and 
mistakes.  The  form  should  represent  a  logical  order  of  work, 
and  should  involve  as  little  repetition  of  figures  as  possible. 
All  scribbling,  multiplying,  dividing,  interpolating,  etc., 
should  be  done  on  separate  sheets  of  paper  from  the  regular 
computation. 

Probable  Errors. 

282.  The  reader  who  does  not  understand  the  principles 
of  least  squares  cannot  hope  to  understand  the  logic  of  the 
formulae  given  in  Chapters  IV  and  V  for  certain  least-square 
computations.     But  after  a  careful  perusal  of  §§  283-285   a 
statement  of  the  uncertainty  in  a  certain  value  in  terms  of 
the  so-called  probable  error  should  not  be  unintelligible  to 
him. 

283.  In    the    expression    "probable    error"    the    word 
"  probable  "  is  not  used  in  its  ordinary  sense,  but  in  a  special 


§  283.  PROBABLE  ERROR.  ?73 

technical  sense.  To  assert  that  the  probable  error  of  a  cer- 
tain stated  value  is  ±  ^,  is  to  assert  the  chances  are  equal  for 
and  against  the  truth  of  the  proposition  that  the  stated  value 
does  not  differ  from  the  truth  by  more  than  e.  Thus,  to 
assert  that  the  azimuth  of  a  certain  line  west  of  north  as 
derived  from  a  certain  series  of  observations  is  59". o  ±  o".5, 
is  to  assert  that  it  is  as  likely  that  the  true  value  of  that 
azimuth  is  between  58". 5  and  59". 5  W.  of  N.  as  that  it  is 
some  value  outside  of  these  limits.  To  assert  that  the  prob- 
able error  of  a  single  observation  in  a  series  =  ±  o".5,  is  to 
assert  that  it  is  an  even  chance  that  any  particular  observation 
is  within  o".5  in  either  direction  of  the  truth.  Or,  what  is 
the  same  thing,  it  is  to  assert  that  if  a  long  series  of  such 
observations  were  made,  the  chances  are  that  one-half  of  the 
observations  would  give  results  within  o".5  of  the  truth,  and 
one-half  would  give  results  differing  from  the  truth  by  more 
than  o".5. 

More  accurately,  perhaps,  the  probable  error  should  be 
regarded  as  referring  to  accidental*  errors  only,  without 
reference  to  possible  constant  errors.  Thus  the  above  state- 
ments should  be  modified  to  read  as  follows:  To  assert  that 
the  azimuth  of  a  certain  line  west  of  north,  as  derived  from  a 
certain  series  of  observations,  is  59". o  ±  0^.5,  is  to  assert  that 
if  an  infinite  number  of  such  observations  were  taken,  under 
the  same  average  conditions,  their  mean  would  be  as  likely  to 
lie  between  58".  5  and  $9".$  W.  of  N.  as  to  fall  outside  those 
limits.  And  to  assert  that  the  probable  error  of  a  single 
observation  =  ±  o".$,  is  to  assert  that  it  is  an  even  chance 
that  that  particular  observation  is  within  o".5  in  either  direc- 
tion of  the  mean  which  would  result  from  an  infinite  number 
of  such  observations,  made  under  the  same  average  condi- 

*  For  the  distinction  between  accidental  and  constant  errors  see  foot- 
note to  §  74. 


274  GEODETIC  ASTRONOMY.  §  286. 

tions.  The  second  form  of  the  statement  is  non-committal 
as  to  possible  constant  errors  affecting  all  the  series  alike, 
which  would  not  be  eliminated  by  increasing  the  number  of 
observations.  Such  a  constant  error  would  be  introduced 
into  an  observed  azimuth  by  placing  the  azimuth  light, 
unknowingly,  a  little  to  one  side  of  the  monument  which  it  is 
supposed  to  indicate. 

284.  There  seems  to  be  some  confusion  between  these  two 
conceptions  of  the  probable  error.      It  is  a  common  mistake 
among  those  who  use  least  squares  to  derive  a  probable  error 
by  methods  which  correspond  to  the  second  form  of  state- 
ment above,  and  then  to  assume  that  the  first  form  of  state- 
ment  is  true.      Hence   one    is    always    on    the  safe  side    to 
assume  the  second  form  of  statement  to  give  the  true  mean- 
ing of  the  probable  error,   and  to  form  an  estimate   of  the 
possibility  of  a  constant  error  from  other  sources  of  informa- 
tion.    If  there   is  no  possibility  of  a  constant  error  in   the 
observations,  the  two  forms  of  statement  are  identical. 

285.  The  relation  between  the  probable  error  of  a  single 
observation,    and   the   total   range   between   the  largest  and 
smallest  values  given  by  such  observations,  is  as  follows:   If 
the  probable  error  of  a  single  observation  is  ±  e,  one  is  to 
expect    that   if  a  large   number  of  such    observations  were 
made,  only  about  one  per  cent  will  fall  outside  a  total  range 
of  7^  times  e.     Or,  if  the  probable  error  of  a  single  observa. 
tion  is  ±  o".5,  only  about  one  observation  in  one  hundred 
would  be  expected  to  fall  outside  a  total  range  of  3 ".8. 

The  Latitude  Variation. 

286.  Until  a  few  years  ago  it  was  supposed  that  the  lati- 
tude of  a  given  station  was  invariable.      During  the  last  few 
years  a  vigorous  investigation  of  that  assumption  has  been 
made,   both  by  means  of  new  series  of  observations  of  the 


§28/.  VARIATION  OF  LATITUDE.  2?$ 

highest  degree  of  accuracy  planned  especially  for  the  purpose, 
and  by  the  re-examination  of  various  old  series  of  observa- 
tions at  the  fixed  observatories.  The  result  of  these  investi- 
gations may  be  briefly  stated  as  follows:  The  axis  of  rotation 
of  the  Earth  does  not  coincide  exactly  with  its  axis  of  figure. 
By  axis  of  figure  is  meant  that  line  about  which  its  moment 
of  inertia  is  a  maximum.  Roughly  speaking,  the  axis  of 
figure  describes  a  cone,  with  its  vertex  at  the  centre  of  the 
Earth,  about  the  axis  of  rotation  once  in  428  days.  The 
motion  of  the  pole  of  figure  about  the  pole  of  rotation  during 
that  interval  is  roughly  an  ellipse  with  a  major  diameter  of 
about  60  feet,  described  in  the  direction  of  decreasing  west 
longitudes,  that  is,  in  a  counter-clockwise  direction,  as  seen 
from  above  at  the  north  pole.  This  motion  is  combined  also 
with  one  of  a  period  of  one  year,  and  is  variable  as  to  the 
diameter  and  position  of  the  ellipse,  and  otherwise,  so  that 
the  above  statement  serves  simply  as  an  approximate  descrip- 
tion of  the  motion.  The  general  law  governing  the  motion 
is  not  yet  known,  and  all  formulae  as  yet  derived  for  predict- 
ing the  future  motion  are  empirical. 

The  direction  of  gravity  at  a  given  station  is  sensibly  con- 
stant as  referred  to  the  axis  of  figure,  that  is,  as  referred  to 
the  solid  Earth.  But  the  latitude  as  measured  is  referred  to 
the  axis  of  rotation — the  plane  perpendicular  to  that  line,  the 
equator,  being  the  plane  to  which  the  declinations  of  the 
stars  are  referred.  Hence  the  latitude  of  every  station  on 
the  Earth  varies  through  a  range  equal  to  twice  the  angle 
between  the  two  axes,  a  range  of  about  o".6.  It  changes 
from  its  maximum  to  its  minimum  value,  and  back  again  to 
the  maximum,  once,  roughly  speaking,  every  428  days. 
This  motion  can  be  traced  in  the  past,  but  not  as  yet  pre- 
dicted for  the  distant  future. 

287.  Three  examples  of  the  long  series  of  latitude  obser- 


276  GEODETIC  ASTRONOMY.  §  288. 

vations  made  with  zenith  telescopes  for  the  special  purpose 
of  determining  the  latitude  variation  may  be  found  in  Coast 
and  Geodetic  Survey  Reports  for  1892,  part  2,  pp.  1-159, 
and  for  1893,  part  2,  pp.  440-508.  The  principal  investi- 
gations of  the  variation  by  means  of  latitude  observations  not 
specially  planned  for  the  purpose  have  been  made  by  Prof. 
S.  C.  Chandler.  Indeed,  his  investigations  first  proved  satis- 
factorily that  such  variations  are  a  fact.  His  results  will  be 
found  published  in  various  numbers  of  the  Astronomical 
Journal  for  several  years  past.  A  general  statement  of  the 
"  Mechanical  interpretation  of  the  variations  of  latitudes," 
by  Prof.  R.  S.  Woodward,  will  be  found  in  the  Astronomical 
Journal  No.  345,  May  21,  1895. 

Station  Errors  and  the  Economics  of  Observing. 

288.  The  author  cannot  close  this  book  without  calling 
attention  briefly  to  one  phase  of  geodetic  astronomy  to  which 
little  attention  has  apparently  been  paid,  but  which  is  of  great 
importance  in  planning  the  astronomical  work  in  connection 
with  a  geodetic  survey,  namely,  the  relation  between  the 
economics  of  observing  and  station  errors. 

Broadly  stated,  the  purpose  of"  the  astronomical  observa- 
tions made  in  connection  with  a  geodetic  survey  is  to  deter- 
mine the  relation  between  the  actual  figure  of  the  Earth  as 
defined  by  the  lines  of  gravity  and  the  assumed  mean  figure 
upon  which  the  geodetic  computations  are  based.*  This  is 
the  purpose,  whether  the  astronomical  observations  be  used 
simply  as  a  check  upon  the  geodetic  operations,  or  whether 
they  be  used  as  a  means  of  determining  the  mean  figure  of 
the  Earth.  In  determining  the  relation  between  the  actual 
figure  and  the  assumed  mean  figure  three  classes  of  errors  are 

*  See  foot-note  to  §  15. 


§  289.  ECONOMICS   OF  OBSERVING. 

encountered:  the  errors  of  the  geodetic  observations;  the 
errors  of  the  astronomical  observations;  and  the  errors  due  to 
the  fact  that  only  a  few  scattered  stations  can  be  occupied  on 
the  large  area  to  be  covered,  and  that  the  station  errors  as 
derived  for  these  few  points  must  be  assumed  to  represent 
the  facts  for  the  whole  area. 

Neglect  the  first  class  of  errors,  as  being  in  the  province 
of  geodesy  rather  than  astronomy.  The  duty  of  the  engineer 
when  planning  the  astronomical  work  of  a  survey  is  to  so  fix 
the  number  and  character  of  the  observations  at  each  station, 
and  the  number  and  position  of  the  stations,  as  to  make  the 
combined  errors  of  the  second  and  third  classes  a  minimum 
for  a  given  expenditure.  By  increasing  the  number  of  obser- 
vations at  a  station  the  errors  of  the  second  class  may  be 
diminished,  the  relation  between  the  number  of  observations 
and  the  error  of  the  result  being  that  said  error  is  inversely 
proportional  to  the  square  of  the  number  of  observations  in 
the  most  favorable  case  (of  no  tendency  to  constant  errors  in 
the  series  of  observations).  If  there  are  any  constant  errors 
affecting  the  series,  then  the  increase  in  accuracy  with  increase 
in  the  number  of  observations  is  slower  than  that  stated 
above.  The  third  class  of  errors  may  be  reduced  by  increas- 
ing the  number  of  stations,  and  distributing  them  as  uniformly 
as  possible,  so  as  to  diminish  the  area  to  which  the  result 
from  each  station  is  assumed  to  apply. 

289.  To  illustrate,  suppose  the  latitude  observations  for 
a  geodetic  survey  of  a  State  are  being  planned.  Let  us  sup- 
pose that  the  engineer  knows  that  with  the  available  zenith 
telescopes  and  star  places  an  observer  can  secure  a  latitude 
with  a  probable  error  of  about  ±  o".  10  from  observations  on 
a  single  evening,  and  that  he  can  reduce  this  to  ±  o".o6  by 
observing  on  four  evenings.  Let  us  assume  that  he  estimates 
that  it  will  cost  the  same,  on  an  average,  to  observe  on  four 


OP  THE 

UNIVERSITY 


2?8  GEODETIC  ASTRONOMY.  §  289. 

nights  at  one  station,  as  to  observe  at  three  stations  on  three 
different  nights.*  Should  he  plan  to  observe  at  ten  different 
stations  distributed  uniformly  over  the  State  on  four  nights 
at  each  station,  or  at  thirty  stations  uniformly  distributed  for 
one  night  only  at  each  ?  Obviously  the  answer  depends 
mainly  on  the  magnitude  of  the  station  errors  to  be  expected. 
If  he  estimates  the  station  error  by  consulting  the  results 
obtained  on  the  U.  S.  Lake  Survey,  he  will  expect  an  average 
station  error  of  nearly  4",  with  a  maximum  exceeding  io".f 
If  he  consults  the  report  J  of  the  "  Survey  of  the  Northern 
Boundary  from  the  Lake  of  the  Woods  to  the  Rocky  Moun- 
tains "  he  finds  that  the  average  station  error  there  was  2" , 
with  a  maximum  of  8",  and  that  in  one  case  six  successive 
stations  on  a  total  distance  of  100  miles  along  the  line  showed 
a  nearly  uniform  change  of  about  o".  14  per  mile  in  one  direc- 
tion. If  he  consults  the  published  results  of  still  other 
surveys  his  estimate  of  the  station  errors  to  be  expected  will 
not  be  materially  altered.  Does  it  not  seem  evident  that 
under  such  circumstances  the  thirty  stations  should  be  occu- 
pied on  one  night  each  ?  Yet  the  usual  practice  of  geodetic 
surveys  in  this  country  corresponds  rather  to  the  plan  of 
observing  at  10  stations  on  4  nights  each,  even  though  the 
observation  error  in  the  result  from  a  single  night  is  upon  an 
average  only  one-twentieth,  say,  of  the  station  error. 

With  longitudes  and  azimuths  it  will  be  found  that  the 
ratio  of  the  errors  of  the  astronomical  observations  to  the 
station  errors  is  somewhat  larger,  but  not  enough  larger  to 
materially  modify  the  above  economic  problem. 

*  It  being  expected  that  the  observations  are  to  be  taken  at  triangula- 
tion  stations  by  the  same  observers  who  measure  the  horizontal  angles  of 
the  triangulation. 

f  See  Professional  Papers  of  the  Corps  of  Engineers  No.  24  (Lake  Sur- 
vey Report),  p.  814. 

\  Plate  opposite  page  267  of  that  report. 


§290. 


CONVERSION   TABLES. 


279 


290.     CONVERSION    OF   MEAN    SOLAR   TIME     INTO    SIDEREAL. 

Correction  to  be  added  to  a  mean  solar  interval  to  obtain  the  corresponding  sidereal  interval. 

(See  §  23.) 


Mean 

^h 

,  h 

_h 

_h 

h 

h 

For 

Solar. 

O 

IB 

2 

3 

4 

5h 

Seconds. 

Om 

0™  00*.  000 

om  og'.  856 

o"1  19*.  713 

o™  29'.  569 

om  39s.  426 

o™  49s.  282 

0* 

o'.ooo 

i 

o  oo  .164 

o  10  .021 

o  19  .877 

o  29  .734 

o  39  -590 

o  49  .447 

i 

o  .003 

2 

o  oo  .329 

o  10  .185 

o  20  .041 

o  29  .898 

0  39  -754 

o  49  .611 

2 

o  .005 

3 

o  oo  .493 

o  10  .349 

o  20  .206 

o  30  .062 

o  39  .919 

o  49  .775 

3 

o  .oo{ 

4 

o  oo  .657 

o  10  .514 

0   20  .370 

o  30  .227 

o  40  .083 

o  49  .939 

4 

0  .Oil 

5 

o  oo  .821 

o  10  .678 

0   20  .534 

o  30  .391 

o  40  .247 

o  50  .104 

5 

o  .014 

6 

o  oo  .986 

o  10  .842 

0   20  .699 

o  30  -555 

o  40  .412 

o  50  .268 

6 

o  .016 

7 

o  01  .150 

0   II  .006 

0   20  .863 

o  30  .719 

o  40  .576 

o  50  .432 

7 

o  .019 

8 

o  01  .314 

o  ii  .171 

O   21  .027 

o  30  .884 

o  40  .740 

o  50  -597 

8 

o  .022 

9 

o  oi  .478 

o  ii  -335 

0   21  .igi 

o  31  .048 

o  40  .904 

o  50  .761 

9 

o  .025 

10 

o  oi  .643 

o  ii  .499 

0   21  .356 

O   31  .212 

o  41  .069 

o  50  .925 

10 

o  .027 

ii 

o  oi  .807 

o  ii  .663 

O   21  .520 

o  31  .376 

o  41  .233 

o  51  .089 

ii 

o  .030 

12 

o  oi  .971 

0   II  .828 

0   21  .684 

o  31  .541 

o  41  .397 

o  51  .254 

12 

o  .03-: 

'3 

o  02  .  1  36 

o  ii  .992 

O   21  .849 

o  31  .705 

o   i  .561 

o  51  .418 

13 

o  .03^ 

14 

o  02  .300 

0   12  .156 

0   22  .013 

o  31  .869 

o   i  .726 

o  51  .582 

J4 

o  .038 

«5 

o  02  .464 

0   12  .32! 

0   22  .177 

o  32  .034 

o   i  .890 

o  51  .746 

15 

o  .041 

16 
*7 

o  02  .628 

0   02  .793 

0   12  .485 
0   12  .649 

0   22  .341 

o  32  .198 

0    2  .054 

o  51  .911 

16 

o  .044 

18 
19 

0   02  .957 
0   03  .121 

0   12  .978 

O   22  .834 

o  32  .691 

0    2  .383 

o   a  -547 

o  52  .404 

18 
19 

o  .052 

20 

o  03  .285 

o  13  .142 

0   22  .998 

o  32  -855 

O    2  .711 

o  52  .568 

20 

o  .055 

21 

o  03  .450 

o  13  .306 

o  23  .163 

o  33  -OI9 

o  42  .876 

o  52  .732 

21 

o  .057 

22 

o  03  .614 

o  *3  -471 

o  23  .327 

o  33  -'83 

o  43  .040 

o  52  .896 

22 

o  .060 

23 

o  03  .778 

o  13  .635 

o  23  .491 

o  33  -348 

o  43  .204 

o  53  .061 

23 

o  .063 

24 

o  03  .943 

o  n  .799 

o  23  .656 

o  33  -512 

o  43  .368 

o  53  -225 

24 

o  .066 

25 

o  04  .107 

o  13  .963 

o  23  .820 

o  33  -676 

0  43  -533 

o  53  -389 

25 

o  .068 

26 

o  04  .271 

o  14  .128 

o  23  .984 

o  33  .841 

o  43  .697 

o  53  -554 

26 

o  .071 

27 

o  04  .435 

o  14  .292 

o  24  .148 

o  34  .005 

o  43  .861 

o  53  -7i8 

27 

o  .074 

28 

o  04  .600 

o  14  .456 

o  24  .313 

o  34  -169 

o  44  .026 

o  53  .882 

28 

o  .077 

29 

o  04  .764 

o  14  .620 

o  24  .477 

o  34  -333 

o  44  .190 

o  54  .046 

29 

o  .079 

3° 

o  04  .928 

o  14  .785 

o  24  .641 

o  34  .498 

0  44  -354 

o  54  .211 

30 

o  .082 

31 

o  05  .093 

o  14  .949 

o  24  .805 

o  34  .662 

o  44  .518 

0  54  -375 

31 

o  .085 

32 

o  05  .257 

o  15  .113 

o  24  .970 

o  34  .826 

o  44  .683 

o  54  -539 

32 

o  .088 

33 

o  05  .421 

o  15  .278 

o  25  .134 

o  34  .990 

o  44  .847 

o  54  .703 

33 

o  .090 

34 

o  05  .587 

o  15  .442 

o  25  .298 

o  35  -155 

o  45  .on 

o  54  .868 

34 

o  .093 

35 

o  05  .750 

o  15  .606 

o  25  .463 

o  35  -3*9 

o  45  .176 

o  55  .032 

o  .096 

36 

o  05  .914 

o  15  .770 

o  25  .627 

o  35  -483 

0  45  -340 

o  55  .196 

36 

o  .099 

37 

o  06  .078 

o  15  -935 

o  25  .791 

o  35  .648 

o  45  .504 

o  55  -361 

37 

O  .  IOI 

38 

o  06  .242 

o  16  .099 

o  25  .955 

o  35  .812 

o  45  .668 

o  55  -525 

38 

o  .104 

39 

o  06  .407 

o  16  .263 

o  26  .120 

o  35  .976 

0  45  -833 

o  55  .689 

39 

o  .107 

40 

o  06  .571 

o  16  .427 

o  26  .284 

o  36  .140 

o  45  -997 

o  55  -853 

40 

0  .110 

41 

o  06  .735 

o  1  6  .592 

o  26  .448 

o  36  .305 

o  46  .161 

o  56  .018 

4i 

O  .  112 

42 

o  06  .900 

o  16  .756 

o  26  .612 

o  36  .469 

o  46  .325 

o  56  .182 

42 

o  .115 

43 

o  07  .064 

o  i  6  .920 

o  26  .777 

o  36  .633 

o  46  .490 

o  56  .346 

43 

o  .118 

44 

o  07  .228 

o  17  .085 

o  26  .941 

o  36  .798 

o  46  .654 

o  56  .510 

44 

o  .120 

9 

o  07  .392 
o  07  .557 

o  17  .249 
o  17  .413 

o  27  .105 
o  27  .270 

o  36  .962 
o  37  .126 

o  46  .818 
o  46  .983 

o  56  .675 
o  56  .839 

45 
46 

0.123 

O  .  1  2O 

47 

o  07  .721 

o  17  .577 

o  27  .434 

o  37  .290 

0  47  -»47 

o  57  .003 

47 

o  .  129 

48 

o  07  .885 

o  17  .742 

o  27  .598 

o  37  -455 

o  47  -3" 

o  57  .  168 

48 

o  .131 

49 

o  08  .049 

o  17  .906 

o  27  .762 

o  37  .619 

0  47  -475 

o  57  -332 

49 

0  -134 

5° 

o  08  .214 

o  18  .070 

o  27  .927 

o  37  -783 

o  47  .640 

o  57  .496 

50 

o  .137 

5i 

o  08  .378 

o  18  .234 

o  28  .091 

o  37  -947 

o  47  .804 

o  57  .660 

5' 

o  .140 

52 

o  08  .542 

o  18  .399 

o  28  .255 

0   38  .112 

o  47  .968 

o  57  -825 

52 

o  .142 

53 
54 

o  08  .707 
o  08  .871 

9  18  .563 
o  18  .727 

o  28  .420 
o  28  .584 

o  38  .276 
o  38  .440 

o  48  .132 
o  48  .297 

o  57  .989 
o  58  .153 

53 
54 

0.145 
0.148 

55 

o  09  .035 

o  18  .892 

o  28  .748 

o  38  .605 

o  48  .461 

o  58  .317 

55 

o  .151 

S^ 

o  09  .199 

o  19  .056 

o  28  .912 

o  38  .769 

o  48  .625 

o  58  .482 

56 

0  .153 

57 

o  09  .364 

o  19  .220 

o  29  .077 

o  38  -933 

o  48  .790 

o  58  .646 

57 

o  .156 

58 

o  09  .528 

o  19  .384 

o  29  .241 

o  39  .097 

o  48  .954 

o  58  .810 

58 

o  .159 

59 

o  09  .692 

o  19  .549 

o  29  .405 

o  39  .262 

o  49  .118 

o  58  -975 

59  o  .162 

Mean 

h 

h 

h 

H 

,h 

h 

For 

Solar. 

on 

I* 

2n 

3 

4 

5 

Seconds. 

280 


GEODETIC  ASTRONOMY. 


§290. 


CONVERSION    OF   MEAN    SOLAR   TIME    INTO    SIDEREAL. 

Correction  to  be  added  to  a  mean  solar  interval  to  obtain  the  corresponding  sidereal  interval. 


Mean 

h 

oh 

T  h 

|j 

For 

Solar. 

7 

o 

9 

IO 

Seconds. 

om 

o"  59'.  139 

-  08'.  995 

m  iS«.852 

»  28".  708 

m  38'  .565 

">  48'.  421 

0" 

o'.ooo 

I 

o  59  '303 

09  .160 

19  .016 

28  .873 

38  .729 

48  .585 

i 

o  .003 

2 

o  59  .467 

09  .324 

19  .186 

29  .037 

38  .893 

48  .750 

2 

o  .005 

3 

o  50  .632 

09  .488 

'9  -345 

29  .201 

39  -058 

48  .914 

3 

o  .008 

4 
5 

o  59  .796 
o  59  .960 

09  .652 
09  .817 

19  -509 
19  -673 

29  -365 
29  -530 

39  .222 

39  -386 

49  -078 
49  -243 

4 

5 

0  .Oil 

o  .014 

6 

oo  .  124 

09  .981 

19  -837 

29  .694 

39  -550 

49  .407 

6 

o  .016 

7 

oo  .289 

10  .145 

20  .002 

29  .858 

39  -715 

49  -571 

7 

o  .019 

8 

oo  .453 

10  .310 

20  .166 

30  .022 

39  -879 

49  -735 

8 

o  .022 

9 

oo  .617 

10  .474 

20  .330 

30  .187 

40  .043 

49  -900 

9 

o  .025 

10 

oo  .782 

10  .638 

20  .495 

30  .351 

40  .207 

50  .064 

10 

o  .027 

ii 

oo  .946 

10  .802 

20  .659 

3°  '5T5 

40  .372 

50  .228 

ii 

o  .030 

12 

0   .110 

10  .967 

20  .823 

30  .680 

40  .536 

So  -393 

12 

o  .033 

13 

o  .274 

II  .131 

20  .987 

30  .844 

40  .700 

50  -557 

13 

o  .036 

14 

o  .439 

II  .295 

21  .152 

31  .008 

40  .865 

50  .721 

14 

o  .038 

15 

o  .603 

ii  -459 

21  .316 

31  .172 

41  .029 

50  .885 

15 

o  .041 

16 

o  .767 

ii  .624 

.480 

3T  -337 

4i  .193 

51  .050 

16 

o  .044 

17 

o  .932 

n  .788 

.644 

31  .501 

4i  -357 

5i  -214 

17 

o  .047 

18 

02  .096 

ii  .952 

.809 

31  -665 

4i  -522 

5i  .378 

18 

o  .049 

ig 

02  .260 

12  .117 

•973 

31  -829 

41  .686 

Si  -542 

i9 

o  .052 

20 

02  .424 

12  .28l 

3i  -994 

41  -850 

51  -707 

20 

o  .055 

21 

02  .589 

12  .445 

.302 

32  .158 

42  .015 

5t  -871 

21 

o  .057 

22 

02  .753 

12  .609 

.466 

32  .332 

42  .179 

52  .035 

22 

o  .060 

23 

02  .917 

12  .774 

.630 

32  -487 

42  .343 

52  .200 

23 

o  .063 

24 

03  .081 

12  .938 

22  -794 

32  .651 

42  .507 

52  .364 

24 

o  .066 

25 

03  .246 

13  .IO2 

22  .959 

32  .815 

42  .672 

52  .528 

25 

o  .068 

26 

03  .410 

I3  .266 

23  .123 

32  -979 

42  .836 

52  .692 

26 

o  .071 

27 

03  .574 

13  '43i 

23  .287 

33  -144 

43  .000 

52  .857 

27 

o  .074 

28 

03  -739 

13  -595 

23  -451 

33  -308 

43  -164 

53  -021 

28 

o  .077 

29 

03  -903 

13  -759 

23  .616 

33  -472 

43  -329 

53  -185 

29 

o  .079 

30 

04  .067 

13  -924 

23  .780 

33  '637 

43  -493 

53  -349 

3° 

o  .082 

31 

04  .231 

14  .088 

23  -944 

33  -801 

43  -657 

53  -5M 

31 

o  .085 

32 

04  .396 

14  -252 

24  .109 

33  -965 

43  -822 

53  -678 

32 

o  .088 

33 

04  .560 

14  .416 

24  -273 

34  -129 

43  -986 

53  -842 

33 

o  .090 

34 

04  .724 

14  081 

24  -437 

34  -294 

44  -150 

54  -007 

34 

o  .093 

04  .888 

14  -745 

24  .601 

34  .458 

44  -314 

54  •I7i 

35 

o  .096 

36 

05  -053 

14  .909 

24  .766 

34  -622 

44  -479 

54  -335 

36 

o  .099 

37 

05  .217 

15  -073 

24  -93° 

34  .786 

44  -643 

54  -499 

37 

O  .  IOI 

38 

05  .381 

15  -238 

25  -°94 

34  -95* 

44  -807 

54  .664 

38 

o  .104 

39 

05  -546 

15  -402 

25  -259 

35  ."5 

44  -971 

54  .828 

39 

o  .  107 

40 

05  .710 

15  -566 

25  -423 

35  -279 

45  -136 

54  .992 

40 

o  .no 

41 

05  .874 

15  -731 

25  -587 

35  -444 

45  .300 

55  -156 

4' 

0  .112 

42 

06  .038 

15  .895 

25  -751 

35  .608 

45  -464 

55  -321 

42 

o  .115 

43 

06  .203 

16  .059 

25  -9«6 

35  '772 

45  -629 

55  -485 

43 

o  .118 

44 

06  .367 

16  .223 

26  .080 

35  -936 

45  -793 

55  -649 

44 

o  .120 

45 
46 

06  .531 

06  .695 

16  .388 
16  .552 

26  .244 
26  .408 

36  .101 
36  .265 

45  -957 

46  .121 

55  -814 
55  -978 

45 
46 

o  .123 
o  .126 

47 

06  .860 

16  .716 

26  .573 

36  .429 

46  .286 

56  .142 

47 

o  .  129 

48 

07  .024 

16  .881 

26  .737 

36  -593 

46  .450 

56  .306 

48 

o  .131 

49 

07  .188 

17  .045 

26  .901 

36  .758 

46  .614 

56  -47' 

49 

o  •134 

So 

07  -353 

17  .209 

27  .066 

36  .922 

46  .778 

56  .635 

5° 

o  -137 

07  -5'7 

17  -373 

27  .230 

37  -086 

46  .943 

56  .799 

51 

o  .140 

52 

07  .681 

17  .538 

27  -394 

37  -251 

47  -107 

56  .964 

52 

o  .142 

53 

07  .845 

17  .702 

27  .558 

37  '415 

47  -271 

57  -128 

53 

o  .145 

54 

08  .010 

17  .866 

27  -723 

37  -579 

47  .436 

57  -292 

54 

o  .148 

08  .174 

18  .030 

27  .887 

37  -743 

47  .600 

57  .456 

55 

o  .151 

56 

08  .338 

18  .195 

28  .051 

37  -908 

47  -764 

57  -621 

56 

58 

08  .502 
08  .667 

'8  -359 
18  .523 

28  .215 
28  .380 

38  .072 
38  .236 

47  -928 
48  .093 

57  -785 
57  -949 

57 
58 

o  .  156 
o  .159 

59 

08  .831 

1  8  .688 

a8  .544 

38  .400 

48  .257 

58  .113 

59 

o  .162 

Mean 
Solar. 

6" 

7h 

8" 

9h 

io* 

ii»» 

For 
Seconds. 

§290. 


CONVERSION   TABLES. 


281 


CONVERSION    OF   MEAN    SOLAR   TIME    INTO   SIDEREAL. 

Correction  to  be  added  to  a  mean  solar  interval  to  obtain  the  corresponding  sidereal  interval. 


Mean 

h 

h 

F 

'or 

Solar. 

I 

if 

H 

15 

16 

17° 

Sec 

onds. 

0™ 

i»  58'.  278 

2mo8».i34 

m  27'.  847 
28  .on 

__  Q/:Q 

2°  47'-  56o 

o" 

o'.ooo 

2 

58  .606 

08  .463 

18  .319 

28  .176 

37  -808 
38  .032 

47  -724 
47  .889 

2 

o  .003 
o  .005 

3 

58  -771 

08  .627 

18  .483 

28  .340 

38  .196 

48  .053 

3 

o  .008 

4 

58  -935 

08  .791 

18  .648 

28  .504 

38  .361 

48  .217 

4 

0  .Oil 

5 

59  -099 

08  .956 

18  .812 

28  .668 

38  025 

48  .381 

5 

o  .014 

6 

59  -263 

09  .  12O 

18  .976 

28  .833 

38  .689 

48  .546 

6 

o  .016 

7 

59  .428 

09  .284 

19  .141 

28  .997 

38  -854 

48  .710 

7 

o  .019 

8 

59  -592 

09  .448 

19  -305 

29  .161 

39  -018 

48  .874 

8 

o  .022 

9 

59  -756 

09  .613 

19  .469 

29  .326 

39  •  182 

49  -039 

9 

o  .025 

10 

59  -920 

09  .777 

19  .633 

29  .490 

39  -346 

49  .203 

10 

o  .027 

it 

12 

oo  .085 
oo  .249 

09  .941 

10  .105 

19  .798 
19  .962 

29  -654 
29  .818 

39  -5" 

39  -675 

49  -367 
49  -S31 

n 

12 

o  .030 
0  -°33 

13 

oo  .413 

10  .270 

20  .126 

29  .983 

39  -839 

49  .696 

13 

o  .036 

14 

oo  .578 

10  .434 

2O  .290 

3°  -!47 

40  .003 

49  .860 

14 

o  .038 

IS 

oo  .742 

10  .598 

20  .455 

30  -3" 

40  .168 

50  .024 

15 

o  .041 

16 

oo  .906 

10  .763 

2O  .619 

30  .476 

4°  -332 

50  .188 

16 

o  .044 

17 

oi  .070 

10  .927 

20  .783 

30  .640 

40  .496 

5°  '353 

17 

o  .047 

18 

oi  .235 

II  .091 

20  .948 

30  .804 

40  .661 

50  .517 

18 

o  .049 

19 

oi  .399 

"  -255 

21  .112 

30  .968 

40  .825 

50  .681 

T9 

o  .052 

20 

oi  .563 

II  .420 

21  .276 

3»  -133 

40  .989 

50  .846 

20 

0  -055 

21 

oi  .727 

II  .584 

21  .440 

3i  -297 

41  -153 

51  .010 

21 

o  .057 

22 

oi  .892 

II  .748 

21  .605 

31  .461 

41  .318 

51  .174 

22 

o  .060 

23 

02  .056 

II  .912 

21  .769 

31  -625 

41  .482 

5i  .338 

23 

o  .063 

24 

02  .220 

12  .077 

21  -933 

31  -79° 

41  -646 

51  -5°3 

24 

o  .066 

25 

02  .385 

12  .241 

22  .098 

3i  -954 

41  .810 

5i  -667 

25 

o  .068 

26 

02  .549 

12  .405 

22  .262 

32  .118 

4i  -975 

51  .831 

26 

o  .071 

27 

02  .713 

12  .570 

22  .426 

32  .283 

42  .139 

51  -995 

27 

o  .074 

28 

02  .877 

12  .734 

22  .590 

32  -447 

42  .303 

52  .160 

28 

o  .077 

29 

03  .042 

12  .898 

22  -755 

32  .611 

42  .468 

52  .324 

29 

o  .079 

3° 

03  .206 

13  .062 

22  .919 

32  -775 

42  .632 

52  .488 

30 

o  .082 

03  .370 

13  -227 

23  .083 

32  .940 

42  .796 

S2  .653 

o  .085 

S2 

03  -534 

13  -391 

23  -247 

33  -104 

42  .960 

52  .817 

32 

o  .088 

33 

03  .699 

13  -555 

23  .412 

33  -268 

43  -125 

52  .981 

33 

o  .090 

34 

03  .863 

13  -720 

23  -576 

33  -433 

43  -289 

53  -MS 

34 

o  .093 

35 

04  .027 

13  -884 

23  -740 

33  -597 

43  -453 

53  -310 

35 

o  .096 

36 

04  .192 

14  .048 

23  -905 

33  -761 

43  -6l7 

53  -474 

36 

o  .099 

37 

04  .356 

14  .212 

24  .069 

33  -925 

43  -782 

S3  -638 

37 

o  .101 

38 

04  .520 

M  -377 

24  -233 

34  090 

43  -946 

53  -803 

38 

o  .104 

39 

04  .684 

14  -54' 

24  -397 

34  -254 

44  '"o 

53  -967 

39 

o  .107 

4° 

04  .849 

M  -70S 

24  -562 

34  -418 

44  -275 

54  -!3i 

40 

o  .no 

41 

05  .013 

14  .869 

24  .726 

34  -582 

44  -439 

54  -295 

41 

0  .112 

42 

°5  -»77 

'5  -034 

24  .890 

34  -747 

44  -603 

54  -460 

42 

o  .115 

43 

05  -342 

15  -198 

25  .054 

34  -9" 

44  -767 

54  -624 

43 

o  .118 

44 

05  .506 

15  .362 

25  .219 

35  -075 

44  -932 

54  -788 

44 

0  .120 

45 

05  .670 

*5  -527 

25  .383 

35  -239 

45  -096 

54  -952 

45 

o  .123 

46 

05  .834 

15  .691 

25  -547 

35  -404 

45  .260 

55  -"7 

46 

o  .126 

47 

05  .999 

15  -855 

25  -712 

35  .568 

45  -425 

55  -281 

47 

o  .129 

48 

06  .163 

16  .019 

25  .876 

35  -732 

45  -589 

55  -445 

48 

o  .131 

49 

06  .327 

16  .184 

26  .040 

35  -897 

45  -753 

55  -610 

49 

o  .134 

5° 

06  .491 

16  .348 

26  .204 

36  .061 

45  -9'7 

55  -774 

5° 

o  .  137 

06  .656 

16  .5!2 

26  .369 

36  .225 

46  .082 

55  .938 

o  .140 

S2 

06  .820 

16  .676 

26  .533 

36  .389 

46  .246 

50  .102 

52 

o  .142 

53 

06  .984 

16  .841 

26  .697 

36  -554 

46  .410 

56  -267 

53 

o  .145 

54 

07  .149 

17  .005 

26  .861 

36  .718 

46  -574 

56  .431 

54 

o  .148 

55 

07  -313 

17  .169 

27  .026 

36  .882 

46  -739 

56  -595 

55 

o  .151 

56 
57 

07  .477 
07  .641 

17  -334 
17  .498 

27  .190 
27  -354 

37  -047 
37  -211 

46  .903 
47  -067 

56  -759 
56  .924 

56 
57 

o  -153 
o  .156 

58 

07  .806 

17  .662 

27  -5*9 

37  -375 

47  -233 

57  .088 

58 

o  .159 

59 

07  .970 

17  .826 

27  .683 

37  -539 

47  -396 

57  -252 

59 

o  .  162 

Mean 

£.h 

F 

'or 

Solar. 

I2h 

i3h 

14 

I5h 

16° 

1?h 

Sec 

onds. 

282 


GEODETIC  ASTRONOMY. 


§290. 


CONVERSION    OF    MEAN    SOLAR   TIME    INTO    SIDEREAL. 
Correction  to  be  added  to  a  mean  solar  interval  to  obtain  the  corresponding  sidereal  interval. 


Mean 

h 

For 

Solar. 

i8h 

19" 

2Oh 

2Ih 

22n 

23h 

Seconds. 

om 

2m  57*  -4^7 

3m07«.273 

3m  17*.  129 

3m  26".  986 

3m36'.842 

3""  46'.  699 

o- 

o'.ooo 

i 

57  -581 

3  °7  -437 

3  17  -294 

3  27  .150 

3  37  -007 

3  46  -863 

i 

o  .003 

2 

57  -745 

3  07  .602 

3  17  .458 

3  27  .315 

3  37  -171 

3  47  ^027 

2 

o  .005 

3 

57  -9°9 

3  °7  -766 

3  17  .622 

3  27  .479 

3  37  -335 

3  47  .i92 

3 

o  .008 

4 

58  -074 

3  °7  -93° 

3  17  -787 

3  27  .643 

3  37  -500 

3  47  .356 

4 

o  .on 

5 

58  .238 

3  08  .094 

3  17  -951 

3  27  .807 

3  37  -664 

3  47  -520 

5 

o  .014 

6 

58  .402 

3  °8  -259 

3  18.115 

3  27  .972 

3  37  -828 

3  47  -685 

6 

o  .016 

7 

58  .566 

3  08  .423 

3  18  .279 

3  28  .136 

3  37  -992 

3  47  -849 

7 

o  .019 

8 

58  .731 

3  °8  .587 

3  18  .444 

3  28  .300 

3  38  .157 

3  48  .013 

8 

o  .022 

9 

58  .895 

3  °8  -75r 

3  18  .608 

3  28  .464 

3  38  -321 

3  48.177 

9 

o  .025 

10 

59  .°59 

3  08  .916 

3  18  .772 

3  28  .629 

3  38  -485 

3  48  .342 

10 

o  .027 

it 

59  -224 

3  09  .080 

2  18  .937 

3  28  .793 

3  38  -649 

3  48  .506 

ii 

o  .030 

12 

59  -388 

3  09  .244 

3  19  -101 

3  28  .957 

3  38  -814 

3  48  .670 

12 

o  .033 

13 

59  -552 

3  09  .409 

3  19  -265 

3  29.122 

3  38  .978 

3  48  .834 

J3 

o  .036 

14 

59  .7J6 

3  09  -573 

3  19  -429 

3  29  .286 

3  39  -142 

3  48  -999 

14 

o  .038 

15 

59  -881 

3  °9  -737 

3  iQ  -594 

3  29  .450 

3  39  -3°7 

3  49  -*63 

15 

o  .041 

16 

3  oo  .045 

3  09  ,901 

3  19  .758 

3  29  .614 

3  39  -471 

3  49  -327 

16 

o  .044 

T7 

3  oo  .209 

3  10  .066 

3  19  -922 

3  29  .779 

3  39  -635 

3  49  -492 

J7 

o  .047 

It 

3  oo  .373 

3  10  .230 

3  20  .086 

3  29  .943 

3  39  -799 

3  49  -656 

18 

o  .049 

19 

3  oo  .538 

3  I0  -394 

3  20  .251 

3  30.107 

3  39  -964 

3  49  -820 

'9 

o  .052 

20 

3  oo  .702 

3  1°  -559 

3  20  .415 

3  30  .271 

3  40  .128 

3  49  -984 

20 

o  .055 

21 

3  oo  .866 

3  I0  -723 

3  20  .579 

3  30  -436 

3  40  .292 

3  50  .149 

21 

o  .057 

22 

3  o  .031 

3  10  .887 

3  20  .744 

3  30  .600 

3  40  .456 

3  50  .313 

22 

o  .060 

23 

3  o  .195 

3  1J  -051 

3  20  .908 

3  30  .764 

3  40  .621 

3  5°  -477 

23 

o  .063 

24 

3  o  -359 

3  ii  .216 

3  21  .072 

3  3°  -929 

3  4°  -785 

3  50  .642 

24 

o  .066 

25 

3  o  -523 

3  «  -380 

3  21  .236 

3  3i  -093 

3  4°  -949 

3  50  .806 

25 

o  .068 

26 

3  o  .688 

3  Ix  -544 

3  21  .401 

3  31  -257 

3  41  •1i4 

3  5°  -970 

26 

o  .071 

27 

3  o  .852 

3  "  -70S 

3  21  .565 

3  3i  -421 

3  4i  -278 

3  5i  .134 

27 

o  .074 

28 

3  02  .016 

3  "  -873 

3  21  .729 

3  3i  -586 

3  4i  .442 

3  5i  -299 

28 

o  .077 

29 

3  02  .181 

3  «  .037 

3  21  .893 

3  3i  -750 

3  41  .606 

3  5i  -463 

29 

o  .079 

30 

3  02  .345 

3  12  .201 

3   22  .058 

3  3i  -9M 

3  4i  -771 

3  5i  -627 

30 

o  .082 

31 

3  02  .509 

3   «  .366 

3   22  .222 

3  32  -078 

3  41  -935 

3  5*  -791 

31 

o  .085 

32 

3  °2  -673 

3  "  -530 

3   22  .386 

3  32  .243 

3  42  .099 

3  5i  .956 

32 

o  .088 

33 

3  02  .838 

3  I2  .694 

3   22  .551 

3  32  -407 

3  43  -264 

3  52  .120 

33 

o  ,090 

34 

3  03  .002 

3  12  .858 

3   22  .715 

3  32  -571 

3  42  .428 

3  S2  .284 

34 

o  .093 

35 

3  03  .166 

3  '3  -023 

3   22  .879 

3  32  .736 

3  42  .592 

3  S2  -449 

35 

o  .096 

36 

3  03  .330 

3  *3  -187 

3  23  .043 

3  32  .9°° 

3  42  .756 

3  52  .613 

36 

o  .099 

37 

3  03  .495 

3  X3  -SSI 

3  23  .208 

3  33  -064 

3  42  .921 

3  52  -777 

37 

O  .  IOI 

38 

3  03  .659 

3  »3  -515 

3  23  .372 

3  33  -228 

3  43  .085 

3  52  .941 

38 

o  .104 

39 

3  03  .823 

3  13  -680 

3  23  .536 

3  33  -393 

3  43  .249 

3  53  •  106 

39 

o  .107 

40 

3  03  .988 

3  13  -844 

3  23  -700 

3  33  -557 

3  43  -4^3 

3  53  -270 

40 

o  .  no 

41 

3  04  .152 

3  14  .008 

3  23  .865 

3  33  -72i 

3  43  -578 

3  53  -434 

4i 

0  .112 

4* 

3  04  .316 

3  14  -173 

3  24  .029 

3  33  -886 

3  43  -742 

3  53  .5Q8 

42 

o  .115 

43 

3  04  .480 

3  M  -337 

3  24  .193 

3  34  -050 

3  43  -9°6 

3  53  -763 

43 

o  .118 

44 

3  04  .645 

3  J4  -501 

3  24  .358 

3  34  -2H 

3  44  -071 

3  53  -927 

44 

0  .120 

45 

3  04  .809 

3  *4  -665 

3  24  .522 

3  34  .378 

3  44  -235 

3  54  '°9l 

45 

o  .123 

46 

3  °4  -973 

3  M  -830 

3  24  .686 

3  34  -543 

3  44  -399 

3  54  .256 

46 

o  .126 

47 

3  05  .  137 

3  »4  -994 

3  24  .850 

3  34  .707 

3  44  -563 

3  54  -420 

47 

o  .129 

48 

3  05  .302 

3  15  -158 

3  25  .015 

3  34  -87* 

3  44  -728 

3  54  -584 

48 

o  .131 

49 

3  05  .466 

3  ^5  -322 

3  25  .179 

3  35  -035 

3  44  -892 

3  54  .748 

49 

o  .134 

50 

3  05  .630 

3  J5  -487 

3  25  .343 

3  35  .200 

3  45  -056 

3  54  -913 

50 

o  .137 

Si 

3  °5  -795 

3  *5  -651 

3  25  -508 

3  35  -364 

3  45  -220 

3  55  -°77 

Si 

o  .140 

52 

3  05  .959 

3  15  -815 

3  25  .672 

3  35  -528 

3  45  .385 

3  55  -24* 

52 

o  .  142 

53 

3  °6  .123 

3  15  -980 

3  25  .836 

3  35  -^93 

3  45  -549 

3  55  -4°5 

53 

o  .145 

54 

3  06  .287 

3  16.144 

3  26  .000 

3  35  -857 

3  45  -713 

3  55  -570 

54 

o  .148 

55 

3  06  .452 

3  16  .308 

3  26  .165 

3  36  .021 

3  45  -878 

3  55  -734 

it 

o  .151 

56 

3  06  .616 

3  16  -472 

3  26  .329 

3  36.185 

3  46  .042 

3  55  -898 

56 

0  .153 

57 

3  06  .780 

3  »6  -637 

3  26  -493 

3  36  -350 

3  46  .206 

3  56  -063 

57 

o  .156 

58 

59 

3  06  .944 
3  07  .109 

3  16  .801 
3  1  6  .965 

3  26  .657 
3  26  .822 

3  36  -5*4 
3  36  .678 

3  46  -370 
3  46  -535 

3  56  -227 
3  56  .391 

58 

59 

o  .159 
o  .  162 

Mean 
Solar. 

18" 

i9h 

20" 

It* 

22" 

23" 

For 

Seconds. 

24  mean  solar  hours  =  24h  o3m  s6§.555  of  sidereal  time. 


291, 


CONVERSION   TABLES. 


283 


291.     CONVERSION    OF   SIDEREAL   TIME    INTO    MEAN     SOLAR. 

Correction  to  be  subtracted  from  a  sidereal  interval  to  obtain  the  corresponding  mean  time 

interval.     (See  §  23.) 


Sid. 

Oh 

Ih 

2h 

,h 

.h 

-h 

For 

3 

4 

5 

Seconds. 

o"» 

om  oo».ooo 

om  09*.  830 

om  i98.  659   o111  29».  489 

0-39'-  3i8 

o™  49*.  148 

o» 

o'.ooo 

i 

2 

o  oo  .164 
o  oo  .328 

o  09  .993 
o  10  .157 

o  19  .823 

o  19  -98.7 

o  29  .653 
o  29  .816 

o  39  .482 
o  39  .646 

o  49  .312 
o  49  -475 

i 

2 

o  .003 
o  .005 

3 

o  oo  .491 

o  10  .321 

0   20  .151 

o  29  .980 

o  39  .810 

o  49  .639 

3 

o  .008 

4 

o  oo  .655 

o  10  .485 

0   20  .314 

o  30  .144 

o  39  -974 

o  49  .803 

4 

0  .Oil 

5 

o  oo  .819 

o  10  .649 

o  20  .478 

o  30  .308 

o  40  .137 

o  49  .967 

5 

o  .014 

6 

7 

o  oo  .983 
o  o  .147 

o  10  .813 
o  10  .976 

0   20  .642 

o  20  .806 

o  30  .472 
o  30  .635 

o  40  .301 
o  40  .465 

o  50  .131 
o  50  .295 

6 
7 

o  .016 
o  .019 

8 

o  o  .311 

o  ii  .140 

0   20  .970 

o  30  ,799 

o  40  .629 

o  50  .458 

8 

o  .022 

9 

o  o  .474 

o  ii  .304 

0   21  .134 

o  30  .963 

o  40  .793 

o  50  .622 

9 

o  .025 

o 

o  o  .638 

0   II  .468 

O   21  .297 

o  31  .127 

o  40  .956 

o  50  .786 

10 

o  .027 

i 

0   0   .  802 

o  ii  .632 

0   21  .461 

o  31  .291 

0   41  .120 

o  50  .950 

ii 

o  .030 

2 

o  o  .966 

o  ii  .795 

0   21  .625 

o  31  -455 

o  41  .284 

o  51  .114 

12 

o  .033 

3 

o  02  .  1  30 

o  ii  .959 

O   21  .789 

o  31  .618 

o  41  .448 

o  51  .278 

13 

o  .035 

4 

0   02  .294 

0   12  .123 

0   21  .953 

o  31  -782 

o  41  .612 

o  51  .441 

14 

o  .038 

5 

0   02  .457 

0   12  .287 

O   22  .H7 

o  31  .946 

o  41  .776 

o  51  .605 

15 

o  .041 

6 

0   02  .621 

0   12  .451 

O   22  .280 

o  32  .no 

o  41  .939 

o  51  .769 

16 

o  .044 

7 

0   02  .785 

0   12  .615 

0   22  .444 

o  32  .274 

o  42  .103 

0  5i  '933 

*7 

o  .046 

8 

0   02  .949 

0   12  .778 

0   22  .608 

o  32  .438 

o  42  .267 

o  52  .097 

18 

o  .049 

9 

o  03  .113 

0   12  .942 

0   22  .772 

o  32  .601 

o  42  .431 

o  52  .260 

19 

.052 

20 

o  03  .277 

o  13  .106 

0   22  .936 

o  32  .765 

o  42  .595 

o  52  .424 

20 

•055 

21 

o  03  .440 

o  13  .270 

o  23  .099 

o  32  .929 

o  42  .759 

o  52  588 

21 

.057 

22 

o  03  .604 

o  13  -434 

o  23  .263 

o  33  -°93 

o  42  .922 

o  52  .752 

22 

.060 

23 

o  03  .768 

o  13  .598 

o  23  .427 

o  33  .257 

o  43  .086 

o  52  .916 

23 

.063 

24 

o  03  .932 

o  13  .761 

o  23  .591 

o  33  =420 

o  43  .250 

o  53  .080 

24 

.066 

25 

o  04  .096 

o  13  .925 

o  23  .755 

o  33  -584 

o  43  -4H 

o  53  -243 

25 

.068 

26 

o  04  .259 

o  14  .089 

o  23  .919 

o  33  .748 

o  43  -578 

o  53  -4«>7 

26 

.071 

27 

o  04  .423 

o  14  .253 

o  24  .082 

o  33  .912 

o  43  .742 

o  53  -571 

27 

.074 

28 

o  04  .587 

o  14  .417 

o  24  .246 

o  34  .076 

o  43  .905 

0  53  -735 

28 

.076 

29 

o  04  .751 

o  14  .581 

o  24  .410 

o  34  .240 

o  44  .069 

o  53  .899 

29 

-079 

30 

o  04  .915 

o  14  .744 

o  24  .574 

o  34  -403 

0  44  -233 

o  54  .063 

30 

082 

31 

o  05  .079 

o  14  .908 

o  24  .738 

0  34  -567 

o  44  -397 

o  54  .226 

3* 

.085 

32 

o  05  .242 

o  15  .072 

o  24  .902 

o  34  -731 

o  44  .561 

o  54  -390 

32 

.087 

33 

o  05  .406 

o  15  .236 

o  25  .065 

o  34  -895 

o  44  .724 

o  54  -554 

33 

.090 

34 

o  05  .570 

o  15  .400 

o  25  .229 

o  35  -059 

o  44  .888 

o  54  .718 

34 

•093 

35 

o  05  .734 

o  15  -563 

o  25  .393 

o  35  .223 

o  45  .052 

o  54  .882 

.096 

36 

o  05  .898 

o  15  .727 

o  25  .557 

o  35  .386 

o  45  .216 

o  55  -046 

36 

.098 

37 

o  06  .062 

o  15  .891 

o  25  .721 

o  35  -55° 

0  45  '38° 

o  55  .209 

37 

.101 

38 

o  06  .225 

o  16  .055 

o  25  .885 

o  35  .714 

o  45  -544 

o  55  -373 

38 

.104 

39 

o  06  .389 

o  16  .219 

o  26  .048 

o  35  .878 

o  45  -707 

o  55  -537 

39 

.106 

4° 

o  06  .553 

o  16  .383 

0   26  .212 

o  36  .042 

o  45  .871 

o  55  .701 

4° 

.109 

41 

o  06  .717 

o  16  .546 

o  26  .376 

o  36  .206 

o  46  .035 

o  55  .865 

41 

.112 

42 

o  06  .881 

o  16  ,710 

o  26  .540 

o  36  .369 

o  46  .199 

o  56  .028 

42 

.115 

43 

o  07  .045 

o  16  .874 

o  26  .  704 

o  36  -533 

o  46  .363 

o  56  .192 

43 

.117 

44 

o  07  ,208 

o  17  .038 

o  26  .867 

o  36  .697 

o  46  .527 

o  56  .356 

44 

.120 

45 

o  07  .372 

0   17  .202 

o  27  .031 

o  36  .861 

o  46  .690 

o  56  .520 

45 

"3 

46 

o  07  .536 

o  17  .366 

o  27  .195 

o  37  .025 

o  46  .854 

o  56  .684 

46 

.126 

47 

o  07  .700 

o  17  .529 

o  27  .359 

o  37  .188 

o  47  .018 

o  56  .848 

47 

.128 

48 

o  07  .864 

o  17  .693 

o  27  .523 

o  37  -352 

o  47  .182 

o  57  .on 

48 

•131 

49 

o  08  .027 

o  17  .857 

o  27  .687 

o  37  -5«6 

o  47  .346 

o  57  -175 

49 

•134 

50 

o  08  .191 

o  18  .021 

o  27  .850 

o  37  .680 

o  47  .510 

o  57  -339 

5° 

•137 

51 

o  08  .355 

o  18  .185 

o  28  .014 

o  37  .844 

o  47  .673 

o  57  -503 

51 

•139 

S2 

o  08  .519 

o  18  .349 

o  28  .178 

o  38  .008 

o  47  -837 

o  57  .667 

52 

.142 

53 

o  08  .683 

o  18  .512 

o  28  .342 

o  38  .171 

o  48  .001 

o  57  .831 

53 

•H5 

54 

o  08  .847 

o  18  .676 

o  28  .506 

o  38  -335 

o  48  .165 

o  57  -994 

54 

.147 

$ 

o  09  .010 
o  09  .174 

o  18  .840 
o  19  .004 

o  28  .670 
o  28  .833 

o  38  .499 
o  38  .663 

o  48  .329 
o  48  .492 

o  58  .158 
o  58  .322 

II 

.150 
•T53 

57 

o  09  .338 

o  19  .168 

o  28  .997 

o  38  .827 

o  48  .656 

o  58  .486 

57 

.156 

58 

o  09  .502 

o  19  .331 

o  29  .  161 

o  38  .991 

o  48  .820 

o  58  .650 

58 

.158 

59 

o  09  .666 

o  19  .495 

o  29  .325 

o  39  .154 

o  48  .984 

o  58  .814 

59 

.  161 

Sid. 

o" 

I* 

2h 

-,h 

4h 

5" 

For 
Seconds. 

284 


GEODETIC  ASTRONOMY. 


§29I. 


CONVERSION    OF    SIDEREAL   TIME    INTO    MEAN    SOLAR. 

Correction  to  be  subtracted  from  a  sidereal  interval  to  obtain  the  corresponding  mean  time 

interval. 


Sid. 

6h 

yh 

8" 

9h 

I0» 

11" 

For 
Seconds. 

Oin 

om  58".  977 

mo8V8o7 

m  iS".636 

ra  28".  466 

»  38'.  296 

m  48".  125 

o« 

o*.ooo 

I 

o  59  .141 

08  .971 

18  .800 

28  .630 

38  .459 

48  .289 

i 

o  .003 

2 

o  59  -3°5 

09  -135 

18  .964 

28  .794 

38  .623 

48  -453 

2 

0  .005 

3 

o  59  .469 

09  .298 

19  .128 

28  .958 

38  .787 

48  .617 

3 

o  .oo£ 

4 

0  59  -633 

09  .462 

19  .292 

29  .121 

38  .951 

48  .780 

4 

O  .Oil 

5 

o  59  .796 

09  .626 

19  .456 

29  .285 

39  -"5 

48  .944 

5 

o  .014 

6 

o  59  .960 

09  .790 

19  .619 

29  .449 

39  -279 

49  .108 

6 

o  .016 

7 

oo  .124 

09  -954 

19  .783 

29  .613 

39  -442 

49  -272 

7 

0  .010 

8 

oo  .288 

i  .118 

19  .947 

29  -777 

39  .606 

49  >4?6 

8 

o  .022 

9 

00  -452 

i  .281 

20  .III 

29  .940 

39  -770 

49  .600 

9 

o  .025 

10 

oo  .616 

i  .445 

20  .275 

30  .  104 

39  -934 

49  -763 

10 

o  .027 

ii 

oo  .779 

i  .609 

20  .439 

30  .268 

40  .098 

49  -927 

ii 

o  .030 

12 

oo  .943 

i  -773 

2O  .602 

30  -432 

40  .261 

50  .09! 

12 

0  .033 

'3 

oi  .107 

1   937 

20  .766 

30  -596 

40  .425 

So  .255 

13 

o  .035 

*4 

oi  .271 

I   .100 

2O  .930 

30  .760 

40  .589 

50  .419 

M 

o  .038 

IS 

oi  -435 

i  .264 

21  .094 

30  .923 

4o  -753 

50  .483 

15 

o  .041 

16 

oi  .599 

i  .428 

21  .258 

3i  -087 

40  .917 

50  .746 

16 

o  .04^ 

J7 

oi  .762 

i  .569 

21  .422 

3*  -251 

41  .081 

50  .910 

J7 

o  .046 

18 

oi  .926 

I  .756 

21  .585 

3i  -415 

41  .244 

Si  -074 

it 

o  .049 

*9 

02  .090 

I  .920 

21  .749 

3i  -579 

41  .408 

5i  -238 

19 

o  .052 

20 

02  .254 

I  .083 

21  .913 

3i  -743 

4i  .572 

51  .402 

20 

o  .055 

21 

02  .418 

I  .247 

22  .077 

31  -906 

4i  .736 

5i  -565 

21 

o  .057 

22 

02  .582 

I  .411 

22  .241 

32  .070 

41  .900 

5i  -729 

22 

o  .o€o 

23 

02  -745 

*  -575 

22  .404 

32  .234 

42  .064 

5i  .893 

23 

o  .06; 

24 

02  .909 

i  -739 

22  .568 

32  .398 

42  .227 

52  .057 

24 

o  .066 

25 

03  .073 

i  .903 

22  .732 

32  .562 

42  .391 

52  .221 

25 

o  .068 

26 

03  -237 

13  .066 

22  .896 

32  .726 

42  .555 

52  -385 

26 

o  .071 

27 

03  .401 

13  .230 

23  .060 

32  .889 

42  .719 

52  .548 

27 

o  .07. 

28 

03  .564 

»3  -394 

23   224 

33  -053 

42  .883 

52  .712 

28 

o  .076 

29 

03  .728 

13  588 

23  -387 

33  -217 

43  -°47 

52  .876 

29 

o  .079 

30 

03  -892 

I3  .722 

23  -55' 

33  -381 

43  .210 

53  -°4o 

30 

o  .082 

31 

04  .056 

13  .886 

23  -715 

33  -545 

43  -374 

53  -204 

31 

o  .085 

32 

04  .220 

14  .049 

23  .879 

33  -708 

43  -538 

53  .368 

32 

o  .087 

33 

04  .384 

14  .213 

24  .043 

33  -872 

43  -702 

53  -531 

33 

o  .090 

34 

04  -547 

T4  -377 

24  .207 

34  -036 

43  -866 

53  -695 

34 

o  093 

35 

04  .711 

14  -541 

24  .370 

34  -200 

44  -029 

53  -859 

35 

o  .096 

36 

04  .875 

14  -7°5 

24  -534 

34  -364 

44  -193 

54  -°23 

36 

o  .098 

37 

05  .039 

14  .868 

24  .698 

34  -528 

44  -357 

54  -187 

37 

o  .101 

38 

05  .203 

15  .032 

24  .862 

34  -691 

44  -521 

54  -351 

38 

O  .  IO^ 

39 

05  -367 

15  .196 

25  .026 

34  .855 

44  -685 

54  .5H 

39 

o  .106 

40 

05  .53° 

15  .360 

25  .190 

35  -019 

44  -849 

54  -678 

40 

o  .109 

41 

05  .694 

J5  -524 

25  '353 

35  -183 

45  .012 

54  -842 

41 

0  .112 

42 

05  -858 

15  .688 

25  -517 

35  -347 

45  -176 

55  .°°6 

42 

o  .115 

43 

06  .022 

15  .851 

25  .681 

35  -5" 

45  -340 

55  «17Q 

43 

o  .  117 

44 

06  .186 

16  .015 

25  -845 

35  -674 

45  -504 

55  -333 

44 

0  .120 

45 

06  .350 

16  .179 

26  .009 

35  -838 

45  -668 

55  -497 

45 

o  .123 

46 

06  .513 

1  6  .343 

26  .172 

36  .002 

45  -832 

55  -661 

46 

0  .  I2<! 

47 

06  .677 

16  .507 

26  .336 

36  .166 

45  .995 

55  -825 

47 

o  .128 

48 

06  .841 

16  .671 

26  .500 

36  .330 

46  .159 

55  -989 

48 

o  .  131 

49 

07  .005 

16  .834 

26  .664 

36  -493 

46  .323 

56  .153 

49 

o  -134 

50 

07  .169 

16  .998 

26  .828 

36  -657 

46  .487 

56  .316 

5° 

o  .137 

51 

°7  -332 

17  .162 

26  .992 

36  .821 

46  .651 

56  .480 

5i 

o  .139 

52 

07  .496 

17  .326 

27  -155 

36  .985 

46  .815 

56  .644 

52 

o  .142 

53 

07  .660 

17  .490 

27  .319 

37  -149 

46  .978 

56  .808 

53 

o  .145 

54 

55 

07  .824 
07  .988 

17  «654 
17  .817 

27  .483 
27  647 

37  -313 
37  -476 

47  -H2 
47  -306 

56  .972 
57  -'36 

54 
55 

o  .147 
o  .150 

56 

08  .152 

17  .981 

27  .811 

37  -640 

47  -47<> 

57  -299 

56 

o  .153 

57 

08  .315 

18  .145 

27  -975 

37  -804 

47  -634 

57  -463 

57 

o  .156 

58 

08  .479 

18  .309 

28  .138 

37  -968 

47  -797 

57  -627 

58 

o  .158 

59 

08  .643 

*8  .473 

28  .302 

38  .132 

47  -961 

57  -791 

59 

o  .161 

Sid. 

6h 

7h 

8h 

9h 

I0h 

ii»> 

For 
Seconds. 

§29I. 


CONVERSION   TABLES. 


28S 


CONVERSION   OF   SIDEREAL  TIME   INTO   MEAN   SOLAR. 

Correction  to  be  subtracted  from  a  sidereal  interval  to  obtain  the  corresponding  mean  time 

interval. 


Sid. 

nt 

I3h 

i4h 

* 

1  6" 

.7" 

For 
Seconds. 

0™ 

I 

m  57§-955 
58  .119 

07  .948 

2m  17*.  614 
I7  .778 

m  27'.  443 
27  .607 

10  37*  -273 
37  -437 

m  4  7*.  102 
47  .266 

o- 

i 

oVooo 
o  .003 

2 

58  .282 

08  .112 

17  .941 

27  .771 

37  -601 

47  -400 

2 

o  .005 

3 

58  .446 

08  .276 

18  .105 

27  -935 

37  .764 

47  -594 

3 

o  .008 

4 

58  .610 

08  .440 

18  .269 

28  .099 

37  .9*8 

47  .758 

4 

O  .Oil 

5 

58  -774 

08  .603 

18  .433 

28  .263 

38  .092 

47  .922 

5 

o  .014 

6 

58  .938 

08  .767 

18  .597 

28  .426 

38  .256 

48  .085 

6 

o  .016 

7 

59  «i01 

08  .931 

18  .761 

28  .590 

38  .420 

48  .249 

7 

o  .019 

8 

59  -265 

09  .095 

18  .924 

28  .754 

38  .584 

48  .413 

8 

0  .022 

9 

59  -429 

09  .259 

19  .088 

28  .918 

38  .747 

48  .577 

9 

o  .025 

10 

59  -593 

09  .423 

19  .252 

29  .082 

38  .911 

48  .741 

10 

o  .027 

ii 

59  -757 

09  .586 

19  .416 

29  -245 

39  -075 

48  .905 

ii 

o  .030 

12 

59  -921 

09  .750 

19  .580 

29  .409 

39  .239 

49  .068 

12 

o  .033 

13 

oo  .084 
oo  .248 

09  .914 

10  .078 

19  -744 
19  .907 

29  .573 
29  -737 

38:3 

49  -232 

49  -396 

13 
14 

o  .035 
o  .038 

I  e 

oo  .412 

10  .242 

20  .071 

29  .901 

39  -730 

49  -560 

o  .041 

16 

oo  .576 

10  .405 

20  .235 

30  .065 

39  .894 

49  .724 

16 

o  .044 

17 

oo  .740 

10  .569 

20  .399 

30  .228 

40  .058 

49  .888 

17 

o  .046 

18 

oo  .904 

1°  -733 

20  .563 

30  .392 

40  .222 

50  .051 

18 

o  .049 

19 

oi  .067 

10  .897 

20  .727 

30  .556 

40  .386 

50  .215 

ig 

o  .052 

20 

oi  .231 

ii  .061 

20  .890 

30  .720 

40  -549 

50  -379 

20 

o  .055 

21 

oi  -395 

ii  .225 

21  -054 

30  .884 

40  .713 

50  .543 

21 

o  .057 

22 

oi  -559 

ii  .388 

21  .218 

31  .048 

40  .877 

50  .707 

22 

o  .060 

23 

oi  .723 

ii  -552 

21  .382 

31  .211 

41  .041 

50  .870 

23 

o  .063 

24 

oi  .887 

ii  .716 

21  .546 

3i  -375 

41  -205 

5i  .034 

24 

o  .066 

25 

02  .050 

ii  .880 

21  .709 

3i  -539 

41  .369 

51  .198 

25 

o  .068 

26 

02  .214 

12  .044 

21  .873 

3i  -7°3 

4i  .532 

51  .362 

26 

o  .071 

27 

02  .378 

12  .208 

22  .037 

3i  .867 

41  .696 

5i  -526 

27 

o  .074 

28 

02  .542 

12  .371 

22  .201 

32  .031 

41  .860 

51  .690 

28 

o  .076 

29 

O2  .706 

12  -535 

22  .365 

32  .194 

42  .024 

5i  .853 

29 

o  .079 

30 

02  .869 

12  .699 

22  .529 

32  -358 

42  .188 

52  .017 

3° 

o  .082 

31 

03  -033 

12  .863 

22  .692 

32  .522 

42  .352 

52  .181 

o  .085 

32 

°3  -!97 

13  -°27 

22  .856 

32  .686 

42  .5!5 

52  -345 

32 

o  .087 

33 

03  .361 

13  .igi 

23  .020 

32  .850 

42  .679 

52  .509 

33 

o  .090 

34 
35 

03  .525 
03  .689 

13  -354 
13  .518 

23  .184 
23  -348 

33  -013 
33  .!77 

42  .8*3 
43  .°°7 

52  .673 
52  .836 

34 
35 

o  .093 
o  .096 

36 

03  .852 

13  .682 

23  .512 

33  .341 

43  «i7i 

53  -00° 

36 

o  .098 

37 

04  .016 

13  .846 

23  -675 

33  -505 

43  -334 

53  -I(54 

37 

o  .101 

38 

04  .180 

14  .010 

23  -839 

33  -669 

43  -498 

53  -328 

38 

o  .104 

39 

04  .344 

T4  -!73 

24  -003     33  .833 

43  -662 

53  -492 

39 

o  .  106 

40 

04  .508 

T4  -337 

24  .167 

33  .966 

43  -826 

53  -656 

4° 

o  .109 

04  .672 

14  .501 

24  -331 

34  .160 

43  -990 

53  -819 

0  .112 

42 

04  -835 

14  .665 

24  -495 

34  -324 

44  .154 

53  -983 

42 

o  .  115 

43 

04  .999 

14  .829 

24  .658 

34  -488 

44  -317 

54  -M7 

43 

o  .117 

44 

05  .163 

14  .993 

24  .822 

34  .652 

44  .481 

54  -3" 

44 

0  .120 

45 

»5  -327 

15  .156 

24  .986 

34  -816 

44  .645 

54  -475 

45 

o  .123 

46 

05  .491 

15  -320 

25  .150 

34  -979 

44  .809 

54  -638 

46 

o  .  126 

47 

05  -655 

15  .484 

25  .314 

35  .143 

44  -973 

54  -802 

47 

o  .128 

48 

05  .818 

15  -648 

25  -477 

35  -307 

45  -137 

54.966 

48 

o  .131 

49 

05  .982 

15  .812 

25  .641 

35  -471 

45  '3°° 

55  .!30 

49 

0  -134 

50 

06  .146 

15  -976 

25  .805 

35  -635 

45  -464 

55  -294 

50 

0  .137 

51 

06  .310 

16  .139 

25  -969 

35  .798 

45  -628 

55  -458 

o  .139 

52 

06  .474 

16  .303 

26  .133 

45  -792 

55  -621 

S2 

o  .  142 

53 

06  .637 

16  .467 

26  .297 

36  .126 

45  -956 

55  -785 

53 

o  .145 

54 

06  .801 

16  .631 

26  ,460 

36  .290 

46  .120 

55  -949 

54 

o  .147 

55 

06  .965 

T<5  -795 

26  .624 

36  -454 

46  .283 

56  .113 

55 

o  .150 

56 
57 

07  .129 
07  .293 

16  -959 

17  .122 

26  .788 
26  .952 

36  .618 
36  .781 

46  .447 
46  .611 

56  .277 
56  .441 

56 
57 

0  .153 

o  .156 

58 

°7  -457 

17  .286 

27  .116 

36  «945 

46  -775 

56  .604 

58 

o  .  158 

59 

07  .620 

17  .450 

27  .280 

37  -109 

46  -939 

56  .768 

59 

o  .  161 

Sid. 

U» 

13h 

14" 

:5h 

,« 

'7h 

For 

S<  conds. 

286 


GEODETIC  ASTRONOMY. 


29I, 


CONVERSION    OF   SIDEREAL   TIME    INTO   MEAN     SOLAR. 

Correction  to  be  subtracted  from  a  sidereal  interval  to  obtain  the  corresponding  mean  time 

interval. 


Sid. 

18" 

i9h 

20h 

2l" 

22h 

23" 

For 

Seconds. 

om 

2m  56*.  932 

3mo6«.762 

3*»  i6».59i 

3-»a6».  42  1 

3">36«.25o 

3m  46".  080 

0« 

o'.ooo 

i 

2 

57  .096 
57  -260 

3  °6  -925 
3  07  .089 

3  l6  -755 
3  16  -9!9 

3  26  .585 
3  26  .748 

3  36  .4H 
3  36  .578 

3  46  -244 
3  46  .407 

2 

o  .003 
o  .005 

3 

57  -424 

3  °7  -253 

3  17  -083 

3  26  .912 

3  36  .742 

3  46  .571 

3 

o  .008 

4 

57  -587 

3  07  -417 

3  i7  -246 

3  27  .076 

3  36  -906 

3  46  '735 

4 

o  .on 

57  .751 

3  07  .581 

3  i7  -4*0 

3  27  -240 

3  37  -069 

3  46  -899 

5 

o  .014 

g 

57  -9*5 

3  °7  -745 

3  17  -574 

3  27  .404 

3  37  .233 

3  47  -063 

6 

o  .016 

7 

58  .079 

3  07  .908 

3  i7  -738 

3  27  .568 

3  37  -397 

3  47  -227 

7 

o  .019 

8 

58  .243 

3  08  .072 

3  17  -902 

3  27  -73i 

3  37  .561 

3  47  '390 

8 

0  .022 

9 

58  .406 

3  08  .236 

3  18  .066 

3  27  .895 

3  37  -725 

3  47  -554 

9 

o  .025 

10 

58  .570 

3  08  .400 

3  1  8  .229 

3  28  .059 

3  37  -889 

3  47  -7'8 

10 

o  .027 

ii 

58  .734 

3  08  .564 

3  18  -393 

3  28  .223 

3  38  .052 

3  47  -882 

ii 

o  .030 

12 

58  .898 

3  08  .728 

3  *8  .557 

3  28  -387 

3  38  .216 

3  48  .046 

12 

o  .033 

13 

59  .062 

3  08  .891 

3  18  .721 

3  28  .550 

3  38  -380 

3  48  .210 

T3 

o  .035 

14 

59  -226 

3  °9  -055 

3  18  .885 

3  28  .714 

3  38  -544 

3  48  -373 

14 

o  .038 

15 

59  -389 

3  °9  -219 

3  19  -049 

3  28  .878 

3  38  .708 

3  48  -537 

15 

o  .041 

16 

59  -553 

3  °9  -383 

3  19  -212 

3  29  .042 

3  38  -871 

3  48  .701 

16 

o  .044 

17 

59  >717 

3  °9  -547 

3  19  -376 

3  29  .206 

3  39  -035 

3  48  -865 

17 

o  .046 

18 

59  .881 

3  09  .710 

3  19  -540 

3  29  -370 

3  39  -'99 

3  49  -029 

18 

o  .049 

*9 

3  oo  .045 

3  09  .874 

3  i9  -7°4 

3  29  .533 

3  39  -363 

3  49  -193 

*9 

o  .052 

20 

3  oo  .209 

3  10  .038 

3  19  -868 

3  29  .697 

3  39  -527 

3  49  '356 

20 

0  -055 

21 

3  oo  .372 

3   10  .202 

3  20  .032 

3  29  .861 

3  39  -691 

3  49  -520 

21 

0  -057 

22 

3  oo  .536 

3  10  .366 

3  20  .195 

3  3o  .025 

3  39  -854 

3  49  -684 

22 

o  .o€o 

23 

3  oo  .700 

3  10  .530 

3  20  .359 

3  30  .189 

3  40  .018 

3  49  -848 

23 

o  .063 

24 

3  oo  .864 

3  10  -693 

3  20  .523 

3  30  -353 

3  40.182 

3  50  .012 

24 

o  .066 

25 

3  01  .028 

3  10  .857 

3  20  .687 

3  30  .5J6 

3  40  .346 

3  5°  .J75 

25 

o  .068 

26 

3  01  .192 

3  ii  .021 

3  20  .851 

3  30  .680 

3  4°  .5™ 

3  50  -339 

26 

o  .071 

27 

3  OI  -355 

3  ii  .185 

3  21  .014 

3  3o  .844 

3  40  .674 

3  50  -503 

27 

o  .074 

28 

3  oi  .5*9 

3  "  -349 

3  21  .178 

%  3  31  .008 

3  40  .837 

3  5°  -667 

28 

o  .076 

29 

3  oi  .683 

3  "  •513 

3  21  .342 

3  3i  -'72 

3  41  .001 

3  50  .831 

29 

o  .079 

3° 

3  oi  .847 

3  ii  .676 

3  21  .506 

3  3i  .336 

3  41  -'OS 

3  50  -995 

30 

o  .082 

31 

3  02  .on 

3  ii  -84° 

3  21  .670 

3  3'  -499 

3  4i  -329 

3  5i  -158 

31 

o  .085 

32 

3   °2  .174 

3  12  .004 

3  21  .834 

3  3i  -663 

3  4i  -493 

3  5i  -322 

32 

o  .087 

33 

3  °2  .338 

3  12  .168 

3  21  .997 

3  3i  -827 

3  41  -657 

3  5i  -486 

33 

o  .090 

34 

3  02  .502 

3  12  .332 

3   22  .l6l 

3  3i  -99* 

3  41  .820 

3  51  -650 

34 

o  .093 

35 

3  02  .666 

3  12  .496 

3  22  ;325 

3  32  .155 

3  4i  -984 

3  5T  -814 

35 

o  .096 

36 

3  02  .830 

3  I2  .659 

3   22  .489 
3   22  .653 

3  32  -318 
3  32  .482 

3  42  .148 

3  5i  -978 

36 

o  .098 

38 

3  02  .994 
3  °3  .»57 

3  12  .823 
3  I2  .987 

3   22  .817 

3  32  .046 

3  42  .476 

3  52  -305 

38 

o  .104 

39 

3  03  .321 

3  13  -151 

3   22  .980 

3  32  .810 

3  42  .639 

3  S2  -469 

39 

o  .  106 

4° 

3  °3  -485 

3  J3  -?15 

3  23  .144 

3  32  -974 

3  42  .803 

3  52  .633 

40 

o  .109 

41 

3  03  .649 

3  13  -478 

3  23  .308 

3  33  -'33 

3  42  -967 

3  52  -797 

4i 

0  .112 

42 

3  03  .813 

3  13  -642 

3  23  .472 

3  33  -301 

3  43  -131 

3  S2  .9°i 

42 

o  .  115 

43 

3  °3  -977 

3  i3  -806 

3  23  .636 

3  33  -465 

3  43  -295 

3  53  .124 

43 

o  .117 

44 

3  04  .140 

3  13  -970 

3  23  .800 

3  33  -°29 

3  43  -459 

3  53  -288 

44 

O  .  I2O 

45 

3  °4  .304 

3  14  -X34 

3  23  .963 

3  33  -793 

3  43  -622 

3  53  -452 

45 

0  .121 

46 

3  04  .468 

3  14  .298 

3  24  .127 

3  33  -957 

3  43  -786 

3  53  -616 

46 

o  .126 

47 

3  04  .632 

3  14  -461 

3  24  .291 

3  34  -"I 

3  43  -950 

3  53  -780 

47 

o  .  128 

48 

3  °4  .796 

3  T4  -6*5 

3  24  .455 

3  34  -284 

3  44  -"4 

3  53  -943 

48 

o  .131 

49 

3  04  .960 

3  14  -789 

3  24  .619 

3  34  -448 

3  44  -278 

3  54  -i°7 

49 

o  .134 

5° 

3  °5  -123 

3  T4  -953 

3  24  .782 

3  34  -612 

3  44  .442 

3  54  -271 

50 

0  .137 

51 

3  °5  .287 

3  15  -"7 

3  24  .946 

3  34  -776 

3  44  -605 

3  54  -435 

5i 

o  .139 

S2 

3  05  .451 

3  15  -281 

3  25  .no 

3  34  -940 

3  44  -769 

3  54  -599 

52 

o  .142 

53 

3  °5  .615 

3  15  -444 

3  25  -274 

3  35  -104 

3  44  -933 

3  54  -763 

53 

0  .145 

54 

3  °5  -779 

3  15  .608 

3  25  .438 

3  35  -267 

3  45  -097 

3  54  -926 

54 

o  .147 

55 

3  °5  .942 

3  15  -772 

3  25  .602 

3  35  .43i 

3  45  .261 

3  55  -090 

55 

o  .150 

56 

3  06  .106 

3  15  .936 

3  25  -7^5 

3  35  -595 

3  45  -425 

3  55  .254 

56 

0  .153 

57 

3  06  .270 

3  16  .100 

3  25  -929 

3  35  -759 

3  45  .588 

3  55  '4i8 

57 

o  .  156 

58 
59 

3  °6  -434 
3  06  -598 

3  16  .264 
3  l6  -427 

3  26  .093 
3  26  .257 

3  35  -923 
3  36  .086 

3  45  -752 
3  45  -9i6 

3  55  -582 
3  55  -746 

58 
59 

o  .158 
o  .161 

Sid. 

18" 

19" 

20" 

2th 

22h 
1 

23h 

For 

Seconds. 

24  sidereal  hours  =  24*  -[3*°  ssVgog]  of  mean  solar  time. 


§29*. 


TABLES. 


287 


292.     CHANGE    PER   YEAR    IN    THE   ANNUAL    PRECESSION    IN 

DECLINATION. 

In  units  of  the  fifth  decimal  place  (unit  =  o".ooooi). 

(da.  m~\ 
——   )  is  the  annual  variation  in  right  ascension.    (See  §  42.) 


dam 

Right  Ascension  («w). 

dt 

oh  com 

oh  20°* 

oh  4om 

jh  oom 

ih  20m 

ih  40m 

2h  oom 

2h  20m 

2h  40m 

0s.  0 

—  9 

-  9 

-  8 

-  8 

-8 

-  8 

-8 

-  7 

-  7 

.O 

9 

21 

34 

46 

58 

69 

80 

91 

IOO 

.  I 

9 

23 

36 

So 

63 

76 

88 

99 

110 

.2 

9 

24 

38 

53 

68 

82 

95 

107 

119 

•  3 

9 

25 

41 

57 

73 

88 

102 

116 

128 

•4 

9 

26 

44 

61 

78 

94 

no 

124 

138 

•  5 

9 

28 

46 

65 

83 

IOO 

117 

132 

147 

.6 

9 

29 

49 

69 

88 

1  06 

124 

141 

156 

•7 

9 

30 

51 

72 

93 

112 

131 

149 

1  66 

.8 

9 

31 

54 

76 

98 

119 

139 

158 

175 

•9 

9 

33 

56 

80 

103 

125 

146 

166 

185 

2  .0 

9 

34 

59 

84 

1  08 

131 

153 

174 

194 

2  .1 

9 

35 

62 

88 

"3 

137 

161 

183 

203 

2  .2 

9 

36 

64 

91 

118 

143 

168 

191 

213 

2  -3 

9 

38 

67 

95 

123 

150 

175 

199 

222 

2  .4 

9 

39 

69 

99 

128 

156 

182 

208 

232 

2  -5 

9 

40 

72 

103 

133 

162 

190 

216 

241 

2  .6 

9 

42 

74 

106 

138 

168 

197 

224 

250 

2  .7 

9 

43 

77 

no 

143 

174 

204 

233 

260 

2  .8 

9 

44 

79 

114 

148 

180 

212 

241 

269 

2  .9 

9 

46 

82 

118 

153 

187 

219 

250 

278 

3  -o 

9 

47 

84 

121 

158 

193 

226 

258 

288 

3  -i 

9 

48 

87 

125 

163 

199 

234 

266 

297 

3  -2 

9 

49 

89 

129 

168 

205 

24I 

275 

306 

3  -3 

9 

51 

92 

133 

173 

211 

248 

283 

316 

3  -4 

9 

52 

94 

136 

178 

218 

255 

291 

325 

3  -5 

9 

53 

97 

140 

183 

224 

263 

300 

335 

3-6 

9 

54 

99 

144 

188 

230 

270 

308 

344 

3  -7 

9 

56 

102 

148 

193 

236 

277 

316 

353 

3.8 

9 

57 

104 

151 

197 

242 

284 

325 

363 

3  -9 

9 

58 

107 

155 

202 

248 

292 

333 

372 

4  .0 

9 

60 

no 

159 

207 

254 

299 

342 

38i 

5  -o 

9 

72 

135 

197 

257 

316 

372 

425 

475 

6  .0 

9 

85 

1  60 

234 

307 

377 

445 

509 

569 

7  -o 

9 

98 

186 

272 

357 

439 

5i8 

593 

663 

8  .0 

-9 

—  no 

—  211 

-310 

-  407 

-  501 

-  591 

-676 

-756 

,2h  oom 

I2h  aom 

i  ah  40"! 

I3h  oo™ 

,3h  20°! 

I3h  40m 

I4h  oom 

I4h  20m 

I4h  40m 

Right  Ascension  (am). 

Change  all  signs  when  using  this  lower  argument. 


288 


GEODETIC  ASTRONOMY. 


§292. 


CHANGE    PER   YEAR   IN    THE    ANNUAL    PRECESSION    IN 
DECLINATION. 

In  units  of  the  fifth  decimal  place  (unit  =  c/'.ooooi). 

The  side  argument  (—r     is  the  annual  variation  in  right  ascension. 


d«m 

Right  Ascension  (aw). 

dt 

3h  oom 

3h  2om 

3h  4°m 

4h  oom 

4h  2om 

4h  -jam 

5h  oom 

5^  2om 

5h  40in 

OS.O 

-6 

-  6 

-  5 

-4 

-4 

-3 

—  2 

—  2 

—  I 

.0 

109 

117 

124 

131 

I36 

140 

143 

145 

I46 

.1 

120 

128 

136 

143 

149 

154 

157 

1  60 

161 

.2 

130 

I4O 

148 

156 

162 

167 

171 

174 

175 

•  3 

140 

151 

160 

168 

I76 

181 

186 

188 

190 

•  4 

ISO 

l62 

172 

181 

189 

195 

200 

203 

204 

.5 

161 

173 

184 

194 

202 

208 

214 

217 

219 

.6 

171 

184 

iq6 

206 

215 

222 

228 

231 

233 

.7 

181 

195 

208 

219 

228 

236 

242 

246 

248 

.8 

192 

206 

220 

232 

242 

250 

256 

260 

262 

•9 

202 

218 

232 

244 

255 

263 

270 

274 

277 

2  .0 

212 

229 

244 

257 

268 

277 

284 

289 

291 

2  .1 

222 

240 

256 

270 

281 

291 

298 

303 

306 

2  .2 

233 

251 

268 

282 

294 

304 

312 

317 

320 

2  -3 

243 

262 

280 

295 

308 

3i8 

326 

332 

335 

2  .4 

254 

274 

292 

307 

321 

332 

340 

346 

349 

2  .5 

264 

285 

304 

320 

334 

346 

355 

361 

364 

2  .6 

274 

296 

316 

333 

347 

359 

369 

375 

379 

2  .7 

284 

3°7 

327 

345 

361 

373 

383 

389 

393 

2  .8 

295 

318 

339 

358 

374 

387 

397 

404 

408 

2  .9 

305 

330 

351 

370 

387 

400 

411 

418 

422 

3  -o 

315 

340 

363 

383 

400 

414 

425 

433 

437 

3  .1 

326 

352 

375 

396 

414 

428 

439 

447 

45i 

3  -2 

336 

363 

387 

408 

426 

441 

453 

461 

466 

3  -3 

346 

374 

399 

421 

440 

455 

467 

475 

480 

3  4 

357 

385 

4ii 

434 

453 

469 

481 

490 

495 

3  -5 

367 

396 

423 

446 

466 

482 

495 

504 

509 

36 

377 

408 

435 

459 

480 

496 

509 

5i8 

524 

3  -7 

388 

419 

447 

472 

493 

5io 

524 

533 

538 

3-8 

398 

430 

459 

484 

506 

524 

538 

547 

553 

3  -9 

408 

441 

47i 

497 

519 

537 

552 

562 

567 

4  .0 

418 

452 

483 

5io 

532 

55i 

566 

576 

582 

5  .0 

522 

564 

602 

636 

665 

688 

707 

720 

727 

6  .0 

625 

676 

722 

762 

797 

825 

847 

863 

872 

7  -o 

728 

787 

841 

888 

929 

962 

989 

1007 

1018 

8  .0 

-831 

-899 

-961 

-  1015 

-  1061 

-  1099 

—  1129 

-  1150 

—  1163 

\^  00 

I5h  2om 

I5h  40m 

i6h  ocm 

16^  2om 

i6h  40111 

17^  oora 

I7h  20m 

I7h  40m 

Right  Ascension  (aw.) 

Change  all  signs  when  using  this  lower  argument. 


TABLES. 


289 


CHANGE    PER   YEAR   IN    THE   ANNUAL   PRECESSION    IN 
DECLINATION. 

In  units  of  the  fifth  decimal  place  (unit  =  o".ooooi). 

(da-m\ 
—  J  is  the  annual  variation  in  right  ascension. 


*m 

Right  , 

\scensior 

1  (am). 

dt 

6h  om 

6h  2om 

6h  4om 

7h  oom 

yh  2om 

7h  40m 

gh  oom 

gh  2om 

8h  40™ 

OS.O 

0 

+  1 

+  2 

+  2 

+  3 

+  4 

+  4 

+  5 

+  6 

.O 

—  146 

—  144 

-  142 

-139 

-  134 

-128 

—  122 

-  H5 

-  106 

.1 

160 

159 

156 

153 

148 

142 

135 

127 

117 

.2 

175 

174 

171 

167 

161 

155 

147 

139 

129 

•3 

190 

188 

185 

181 

175 

168 

1  60 

150 

140 

•4 

204 

203 

2OO 

195 

189 

181 

172 

162 

151 

•  5 

219 

217 

214 

209 

202 

195 

I85 

174 

162 

.6 

233 

232 

228 

223 

216 

208 

I98 

1  86 

173 

•7 

248 

246 

243 

237 

230 

221 

210 

198 

184 

.8 

262 

261 

257 

251 

244 

234 

223 

210 

196 

•9 

277 

275 

271 

265 

257 

247 

236 

222 

207 

2  .0 

292 

290 

286 

280 

271 

26l 

248 

234 

218 

2  .1 

306 

304 

300 

294 

285 

274 

26l 

246 

229 

2  .2 

321 

319 

315 

308 

298 

287 

274 

258 

240 

2  -3 

335 

333 

329 

322 

312 

300 

286 

270 

251 

2  .4 

350 

348 

343 

336 

326 

314 

299 

282 

263 

2  -5 

364 

362 

358 

350 

340 

327 

3" 

294 

274 

2  .6 

379 

377 

372 

364 

353 

340 

324 

306 

285 

2  -7 

394 

39i 

386 

378 

367 

353 

337 

318 

296 

2  .8 

408 

406 

401 

392 

38i 

366 

349 

330 

308 

2  -9 

423 

420 

415 

406 

394 

380 

362 

342 

319 

3  -o 

437 

435 

429 

420 

408 

393 

375 

354 

330 

3  -i 

452 

450 

444 

434 

422 

406 

387 

366 

34i 

3  -2 

467 

464 

458 

449 

435 

419 

400 

378 

352 

3  -3 

481 

478 

472 

463 

449 

432 

412 

389 

363 

3  -4 

496 

493 

487 

477 

463 

446 

425 

401 

374 

3  -5 

5io 

508 

501 

491 

477 

459 

438 

413 

385 

3-6 

525 

522 

515 

505 

490 

472 

450 

425 

397 

3  -7 

540 

537 

530 

519 

504 

485 

463 

437 

408 

3  -8 

554 

55i 

544 

533 

5i8 

499 

476 

449 

419 

3  -9 

569 

566 

559 

547 

531 

512 

488 

461 

430 

4  .0 

583 

580 

573 

56i 

545 

525 

5oi 

473 

441 

5  -o 

729 

726 

717 

702 

682 

657 

627 

593 

553 

6  .0 

875 

871 

860 

843 

819 

790 

754 

712 

665 

7  .0 

1021 

1016 

1004 

984 

956 

922 

880 

831 

776 

8  .0 

-  1166 

—  1161 

—  114? 

—  1125 

—  1093 

-  1054 

—  1006 

-951 

-  888 

i8fe  oom 

IS*  2on> 

i8h  40^ 

I9h  oom 

igh  2om 

igh  4o"i 

2oh  oom 

2oh  2om 

2oh  4001 

Right 

Ascensio 

1  («>*). 

Change  all  signs  when  using  this  lower  argument. 


290 


GEODETIC  ASTRONOMY. 


§292. 


CHANGE    PER    YEAR   IN    THE   ANNUAL   PRECESSION    IN 
DECLINATION. 

In  units  of  the  fifth  decimal  place  (unit  =  o".ooooi). 

fd<Lm  X 

The  side  argument  (  — — )  is  the  annual  variation  in  right  ascension. 


^n 

Right  Ascension  (aw). 

dt 

9h  oom 

9h  2om 

gh  40m 

I0h  oom 

lot  2om 

i  oh  4om 

uh  oom 

lit  aom 

lit  40m 

0s.  0 

+  6 

+  7 

+  7 

+  8 

+  8 

+  8 

+  8 

+  8 

+  9 

.0 

-97 

-87 

-76 

-65 

-54 

-42 

-29 

-  17 

—  4 

.1 

107 

96 

85 

73 

60 

47 

33 

19 

5 

.2 

118 

106 

93 

80 

66 

52 

37 

22 

7 

•  3 

128 

H5 

102 

87 

72 

57 

41 

24 

8  • 

•4 

138 

125 

no 

95 

79 

62 

45 

27 

9 

•  5 

149 

134 

118 

102 

85 

67 

48 

29 

10 

.6 

159 

143 

127 

I09 

9i 

72 

52 

32 

12 

•7 

169 

153 

135 

116 

97 

77 

56 

34 

13 

.8 

180 

162 

143 

124 

103 

82 

60 

37 

14 

i  .9 

190 

172 

152 

131 

109 

87 

63 

39 

16 

2  .O 

200 

181 

160 

138 

H5 

92 

67 

42 

17 

2  .1 

210 

190 

168 

146 

122 

97 

7i 

45 

18 

2  .2 

221 

200 

177 

153 

128 

102 

75 

47 

J9 

2  -3 

231 

209 

185 

1  60 

134 

107 

78 

50 

21 

2  -4 

241 

218 

193 

167 

140 

112 

82 

52 

22 

2  -5 

252 

228 

202 

175 

146 

117 

86 

55 

23 

2  .6 

262 

237 

2IO 

182 

152 

122 

90 

57 

24 

2.7 

272 

246 

219 

189 

159 

127 

94 

60 

26 

2  .8 

283 

256 

227 

197 

165 

132 

97 

62 

27 

2  -Q 

293 

265 

235 

204 

171 

137 

101 

65 

28 

3  -o 

303 

274 

244 

211 

177 

142 

105 

67 

30 

3  -i 

314 

284 

252 

218 

183 

147 

109 

70 

31 

3  -2 

324 

293 

260 

226 

190 

152 

112 

72 

32 

3  -3 

334 

303 

269 

233 

196 

157 

116 

75 

33 

3  -4 

345 

312 

277 

240 

202 

162 

1  20 

77 

35 

3  -5 

355 

322 

286 

248 

208 

167 

124 

80 

36 

3-6 

365 

331 

294 

255 

214 

171 

127 

83 

37 

3  -7 

375 

340 

302 

262 

220 

176 

131 

85 

38 

3-8 

386 

350 

311 

269 

226 

181 

135 

88 

40 

3  -9 

396 

359 

319 

277 

233 

187 

139 

90 

4i 

4  .0 

406 

368 

327 

284 

239 

191 

142 

93 

42 

5  -o 

509 

462 

411 

357 

300 

241 

180 

118 

55 

6.0 

612 

555 

494 

430 

362 

291 

218 

143 

68 

7  -o 

715 

649 

578 

503 

424 

341 

256 

169 

80 

8  .0 

—  819 

-743 

-662 

-576 

-485 

-391 

-293 

-  194 

-93 

2ih  oom 

2ih  2om 

2ih  4om 

22^  oom 

22^  2Om 

22h  4om 

23!!  oom 

23!!  20m 

23h  40m 

Right  Ascension  (am). 

Change  all  signs  when  using  this  lower  argument. 


§293. 


TABLES. 


29I 


293.     THE    PARALLAX  OF   THE   SUN  (p)  FOR    THE    FIRST  DAY 
OF    EACH    MONTH.     (See  §66.) 


Altitude. 

« 

• 

i 

to  tn 

•88 

£Q 

oj  tn 
II 

fa 

Iti 

<o 

tS2 

II 

%i 

It 

2 
j>» 

"5 

l—» 

V 

fll 

J5 

0° 

9".o 

9".o 

8".  9 

8".  9 

8".  8 

8".  7 

8".  7 

90° 

3 

9  .0 

9  .0 

8  .9 

8  .8 

8  .8 

8  .7 

8  .7 

87 

6 

9  .0 

8  .9 

8  .9 

8  .8 

8  .7 

8  .7 

8  .7 

84 

9 

8  .9 

8  .9 

8  .8 

8  .8 

8  .7 

8  .6 

8  .6 

81 

12 

8  .8 

8  .8 

8  .7 

8  .7 

8  .6 

8  .5 

8  -5 

78 

15 

8  .7 

8  .7 

8  .6 

8  .6 

8  -5 

8  .4 

8  .4 

75 

18 

8  .6 

8  .6 

8  -5 

8  .4 

8  .4 

8  -3 

8  .3 

72 

21 

8  .4 

8  .4 

8  -3 

8  .3 

8  .2 

8  .2 

8  .1 

69 

24 

8  .2 

8  .2 

8  .2 

8  .1 

8  .0 

8  .0 

8  .0 

66 

27 

8  .0 

8  .0 

8  .0 

7  -9 

7  -8 

7  -8 

7  -8 

63 

30 

7  -8 

7  -8 

7  -7 

7  -7 

7  -6 

7  .6 

7  -6 

60 

33 

7  -6 

7  -5 

7  -5 

7  -4 

7  -4 

7  -3 

7  -3 

57 

36 

7  -3 

7  -3 

7  -2 

7  -2 

7  -i 

7  -i 

7  .0 

54 

39 

7  -o 

7  -o 

6  .9 

6  .9 

6  .8 

6  .8 

6  .8 

5i 

42 

6  .7 

6  .7 

6  .6 

6  .6 

6  .5 

6  .5 

6  -5 

48 

44 

6  -5 

6  -5 

6  .4 

6  .4 

6  -3 

6  -3 

6  -3 

46 

46 

6  -3 

6  .2 

6  .2 

6  .2 

6  .1 

6  .1 

6  .0 

44 

48 

6  .0 

6  .0 

6  .0 

5  -9 

5  -9 

5  -8 

5  -8 

42 

50 

5  -8 

5  -8 

5  -7 

5  -7 

5  -6 

5  -6 

5  -6 

40 

52 

5  -6 

5  -5 

5  -5 

5  -4 

5  -4 

5  -4 

5  -4 

38 

54 

5  -3 

5  -3 

5  -2 

5  -2 

5  -2 

5  -i 

5  -r 

36 

56 

5  -o 

5  -o 

5  -o 

5  -o 

4  -9 

4  .9 

4  .9 

34 

58 

4  .8 

4  .8 

4  -7 

4  -7 

4  -7 

4  .6 

4  .6 

32 

60 

4  -5 

4  -5 

4  -5 

4  -4 

4  -4 

4  -4 

4  -4 

30 

62 

4  .2 

4  .2 

4  .2 

4  .2 

4  -i 

4  -i 

4  -i 

28 

64 

4  .0 

3  -9 

3  -9 

3  -9 

3  -8 

3  -8 

3  -8 

26 

66 

3  -7 

3  -7 

3  -6 

3  -6 

3  -6 

3  -6 

3  -5 

24 

68 

3  -4 

3  -4 

3  -4 

3  -3 

3  -3 

3  -3 

3  -3 

22 

70 

3  -i 

3  -i 

3  -i 

3  -o 

3  -o 

3  -o 

3  -o 

20 

72 

2  .8 

2  .8 

2  .8 

2  -7 

2  -7 

2  -7 

2  -7 

18 

74 

2  -5 

2  -5 

2  -5 

2  .4 

2  .4 

2  .4 

2  -4 

16 

76 

2  .2 

2  .2 

2  .2 

2  .1 

2  .1 

2  .1 

2  .1 

14 

78 

I  .q 

I  .9 

I  .9 

I  .8 

i  .8 

i  .8 

i  .8 

12 

80 

I  .6 

I  .6 

I  .6 

i  -5 

i  -5 

i  -5 

i  -5 

10 

82 

I  .2 

I  .2 

I  .2 

I  .2 

I  .2 

I  .2 

I  .2 

8 

84 

o  .9 

o  .9 

o  .9 

o  .9 

o  .9 

o  .9 

o  .9 

6 

86 

o  .6 

o  .6 

o  .6 

o  .6 

o  .6 

o  .6 

o  .6 

4 

88 

o  .3 

o  -3 

o  -3 

o  -3 

o  -3 

o  -3 

o  -3 

2 

90 

0  .0 

0  .0 

o  .0 

0  .0 

0  .0 

0  .0 

0  .0 

0 

292 


GEODETIC  ASTRONOMY. 


§294. 


294.     MEAN  REFRACTION  (&M)  BAROMETER  760  MILLIMETERS 

=  29.9  INCHES. 
Temperature  10°  C.  =  50°  F.    (See  §  68.) 


Altitude. 

Mean 
Refraction. 

&«• 
»! 

£% 
u 

Altitude. 

Mean 
Refraction. 

Change  per 
Minute. 

Altitude. 

Mean 
Refraction. 

£« 
\\ 

£2 
u 

Altitude. 

Mean 
Refraction. 

Change  per 
Minute. 

o°oo 

34'  08". 

n".66 

7°oo 

7'  24". 

o".95 

19°  oo 

'  47"-6 

o".i 

33°  oo 

'  29". 

o".o6 

10 

32  15  -9 

10  .88 

10 

7  14  -9 

o  .91 

20 

44  '6 

0  .1 

20 

28  . 

o  .06 

20 

30  31  • 

10  .10 

20 

7  06  .0 

o  .88 

40 

41  .6 

O  .  Ij 

40 

27  • 

o  .05 

3° 

28  53  -9 

9  -64 

3° 

6  57  -4 

o  .84 

20   00 

38  -7 

O  .1* 

34  oo 

26  . 

o  .05 

40 

27  18  .2 

9  .20 

40 

6  49  .1 

o  .81 

20 

35  -9 

0  .!< 

20 

25  .0 

o  .05 

5° 

25  49  .8 

8  .50 

50 

6  41  .2 

o  .78 

40 

33  -2 

o  .i; 

40 

24  .0 

o  .0; 

I  00 

24  28  .3 

7  .82 

8  oo 

6  33  -5 

o  .76 

21   00 

30  .6 

O    .1; 

35  oo 

23  .0 

o  .05 

10 

23  13  -5 

7  -!7 

10 

6  26  .0 

o  .73 

2O 

28  . 

o  .  i; 

20 

22   .0 

o  .0; 

20 

22  04  .9 

6  .58 

20 

6  18  .9 

o  .70 

4° 

2  25  .6 

0  .12 

40 

I   21   .O 

o  .05 

3° 

21  01  .8 

6  .06 

3° 

6   12  .0 

o  .68 

22   00 

2   23  .2 

0  .12 

36  oo 

I   20  .0 

o  .05 

40 

20  03  .7 

5  -60 

40 

6  05  .3 

o  .66 

2O 

2  2O  .9 

0  .12 

30 

i  18  .5 

o  .05 

So 

9  09  .8 

5  -20 

50 

5  58  -9 

o  .63 

40 

2  18  .6 

O  .  II 

37  oo 

I   17  . 

0  .0, 

2  00 

8  19  .7 

4  -84 

9  oo 

5  52  .7 

o  .61 

23  oo 

2   16  .4 

0  .11 

30 

I   IS  -7 

o  .04 

10 

7  33  -I 

4  -So 

20 

5  40  .8 

o  .58 

20 

2   I4  .2 

O  .11 

38  oo 

I   14  .4 

o  .0. 

20 

6  49  .7 

4  .18 

40 

5  29  .7 

o  -54 

40 

2   12  .1 

0  .10 

30 

I   13  -I 

O  ,O< 

3° 

6  09  .5 

3  -88 

10  00 

19  .2 

o  .51 

24  oo 

2   IO  .1 

0  .10 

39  oo 

I   II   .i 

O  .04 

40 

5  32  .1 

3  -62 

20 

09  .4 

o  .48 

20 

2  08  .1 

o  .10 

30 

I   IO  .5 

0  .0, 

5° 

4  57  -i 

3  -39 

40 

00  .1 

o  .46 

40 

2  06  .1 

0  .10 

40  oo 

I  09  .3 

0  .0. 

3  oo 

4  24  .3 

3  -18 

II  00 

51  -2 

o  -43 

25  oo 

04  .2 

o  .09 

30 

I  08  .1 

o  .04 

10 

3  53  -6 

2  .98 

20 

42  .8 

o  .40 

20 

02   .4 

o  .09 

41  oo 

I  06  .9 

0  .0. 

20 

3  24  .8 

2  .79 

40 

35  -o 

o  .38 

40 

oo  .6 

O  .O( 

30 

I  05  .7 

0  .0,, 

30 

2  57  -8 

2  .6l 

2  00 

27  -5 

o  -37 

26  oo 

58  .8 

o  .09 

42  oo 

i  04  .6 

o  .04 

40 

2  32  -5 

2   .46 

20 

20  .3 

o  -35 

20 

57   i 

o  .09 

30 

03  -5 

0   .04 

50 

2  08  .7 

2  -33 

40 

13  -5 

0  -33 

40 

55  -4 

o  .08 

43  oo 

02  .i 

0  .0* 

4  oo 

I  46   .0 

2  .20 

3  oo 

07  .1 

o  .32 

27  oo 

53  -8 

o  .08 

3° 

01  .3 

o  .04 

10 

i  24  .6 

2  .09 

20 

oo  .9 

o  .30 

20 

52   .2 

o  .08 

44  oo 

OO   .2 

o  .03 

20 

I  04   o2 

I  .98 

40 

55  -1 

o  .28 

40 

50  .6 

o  .08 

30 

59  -2 

o  .0; 

3« 

o  44  .9 

i  .88 

4  oo 

49  -5 

o  .27 

28  oo 

49  -i 

o  .08 

45  oo 

S8   .2 

o  .03 

40 

o  26  .5 

i  .79 

20 

44  -2 

o  .26 

20 

47  -6 

o  .07 

30 

57  -2 

o  .03 

50 

o  09  .1 

i  .70 

40 

39  •' 

o  .25 

40 

46  .1 

o  .07 

46  oo 

56  .2 

o  .03 

5  oo 

9  52  .6 

i  .61 

5  oo 

34  .1 

o  .24 

29  oo 

44  -6 

o  .07 

30 

55  .2 

o  .03 

10 

9  36  -9 

i  -54 

20 

29  -4 

o  .23 

20 

43  -2 

o  .07 

47  oo 

54  -2 

o  .03 

20 

9  21  .9 

i  .46 

40 

24  .8 

o  .23 

40 

41  .8 

o  .07 

30 

53  -3 

o  .03 

3° 

9  07  .6 

i  .40 

6  oo 

20  .4 

O  .22 

30  oo 

40  .5 

o  .07 

48  oo 

52  -5 

o  .03 

40 

8  54  .0 

i  -33 

20 

16  .1 

O  .21 

20 

39  -i 

o  .07 

30 

51  .6 

o  .03 

5° 

8  41  .0 

i  .27 

40 

12  .0 

0  .20 

40 

37  -8 

o  .06 

49  oo 

50  .7 

o  .03 

6  oo 

10 

8  28  .6 
8  16  .7 

I   .22 
I   .16 

7  oo 
20 

08  .2 

04  -5 

o  .19 
o  .19 

31  oo 

20 

36  .6 
35  -3 

o  .06 
o  .06 

30 
50  oo 

49  -8 
48  .9 

o  .03 
o  .03 

20 

8  05  .3 

I   .12 

40 

oo  .9 

o  .18 

40 

34  •> 

o  06 

30 

48  .0 

o  .03 

30 

7  54  -3 

I   .07 

8  oo 

57  -4 

o  .17 

32  oo 

32  .0 

o  .06 

51  oo 

47  .2 

o  .03 

40 

7  43  -9 

I   .02 

20 

54  -o 

o  .17 

20 

31  .8 

o  .06 

30 

46  -3 

o  .03 

5° 

7  33  -9 

o  .98 

40 

5«  -7 

o  .16 

40 

30  -6 

o  06 

52  oo 

45  -5 

o  .03 

§295. 


TABLES. 


293 


MEAN   REFRAC- 
TION 


295.  CORRECTION  TO  MEAN  REFRACTION 
AS  GIVEN  IN  §294,  DEPENDING  UPON 
THE  READING  OF  THE  BAROMETER. 

R  =  (RM^CB)(CD}(CA).    (See  §  68.) 


Altitude. 

Mean 
Refraction. 

1 

&«; 

11 

ss 

52°  30' 
53  oo 
30 

o'  44"- 
o  43  - 
o  43  • 

o".03 
o  .03 
o  .03 

54  oo 
3° 
55  oo 

o  42  . 
o  41  . 
o  40  . 

o  .03 
o  .03 
o  .03 

,  30 
56  oo 
57  oo 

o  40  . 
0  39  • 
o  37  • 

o  .03 
o  .025 
o  .02, 

58  oo 
59  oo 
60  oo 

o  36  -4 
o  35  -o 
o  33  .6 

o  .02; 
o  .02; 

0  .022 

61  oo 
62  oo 

63  oo 

o  32  .3 
o  31  .0 
o  29  .7 

0  .022 

o  .022 

0  .022 

64  oo 
65  oo 
66  oo 

o  28  .4 
o  27  .2 
o  25  .9 

O  ,O2I 
0  .021 
O  .O2I 

67  oo 
68  oo 
69  oo 

o  24  .7 
o  23  .6 

0  22  .4 

o  .020 
o  .020 

0  .020 

70  oo 
71  oo 
72  oo 

0  21   .2 
0  20  .1 

o  18  .9 

o  .019 
o  .019 
o  .019 

73  o° 
74  oo 
75  oo 

o  17  .8 
o  16  .7 
o  15  .6 

o  .018 
o  .018 
o  .018 

76  oo 
77  oo 
78  oo 

o  14  -5 
o  !3  -5 

0   12   .4 

o  .018 
o  .018 
o  .018 

79  oo 
80  oo 
81  oo 

0   II   .3 
0  10  .3 
0  09  .2 

o  .018 
o  .018 
o  .018 

82  oo 
83  oo 
84  oo 

08   .2 
07   .2 
06  .1 

o  .018 
o  .018 
o  .018 

85  oo 
86  oo 
87  oo 

05   .1 
04   .1 
03   .1 

o  .018 
o  .017 
o  .017 

88  oo 
89  oo 
90  oo 

02   .0 
01   .0 

oo  .0 

o  .017 
o  .017 
o  .017 

. 

u'S 

ai 

.£ 

•£j 

4J  +•» 

*-> 

if 

8 

O  w 

Barome 
Inches, 

11 
85 

CB 

*J   00 

$a 

|| 

CB 

Baromet 
Inches. 

Baromet 

Millime 

CB 

20.  o 
20.  i 

20.2 

508 

5" 

0.67 
0.67 
0.67 

24.2 

24-3 
24.4 

615 
6i7 
620 

0.809 

0.81 
0.81 

28.4 
28.5 
28.6 

721 
724 

726 

0.949 

0-953 
0.956 

20.3 
20.4 
20.5 

5« 

0.67 

0.68 
0.68 

24-5 
24.6 
24.7 

622 

625 
627 

0.82 
0.82 
0.82 

28.7 
28.8 
28.9 

729 
732 
734 

0-959 
0.963 
0.966 

20.6 

20.7 

20.8 

I 

0.688 
0.692 
0.696 

24.8 

24.9 
25.0 

63o 
632 
635 

0.82 
0.83 
0.83 

19.  o 
29.1 
29.2 

737 
739 
742 

0.970 
0.973 
0.976 

20.9 

21.0 

53* 
533 

0.699 
0.703 

25-1 
25.2 

637 
640 

0.83 
0.84 

29-3 
29-4 

744 
747 

0-979 
0.983 

21.  I 

536 

0.706 

25-3 

643 

0.84 

29-5 

749 

0.986 

21.2 
21.3 
21.4 

538 
544 

0.709 
0.712 
0.716 

S.I 

645 
648 
650 

0.84 
0.85 
0.85 

29.6 
29.7 
29.8 

752 
754 
757 

0.989 
0.992 
0.996 

21-5 
21.6 

546 
549 

0.719 
0.722 

25-7 
25-8 

653 
655 

0.85 
0.86 

29-9 
30.0 

759 
762 

0.999 
-1.003 

21.7 

55i 

0.725 

25-9 

658 

0.86 

30.1 

765 

i  .007 

21.  S 
21.9 
22.0 

554 
556 
559 

0.729 
0.732 
0-735 

26.0 
26.1 
26.2 

660 
663 
665 

0.869 
0.87 
0.875 

30.2 
30.3 
30.4 

767 
770 
772 

1.  010 

1.013 
1.016 

22.1 
22.2 
22.3 

1 

o-739 
0.742 
0.746 

26.3 
26.4 
26.5 

668 
671 
673 

0.879 
0.882 
0.885 

30.5 
30.6 
30-7 

775 
777 
780 

1.020 

1.023 

1.026 

22.4 
22.5 
22.6 

569 
572 
574 

0.749 
0.752 
0-755 

26.6 
26.7 
26.8 

676 
678 

68  1 

0.889 
0.892 
0.896 

30.8 
30.9 

782 
785 
787 

1.029 
1  -°33 
1.036 

22.7 

576 

0.759 

26.O 

683 

0.800 

22.8 

579 

0.765 

27.0 

686 

v.  wyv 

0.902 

22.9 

582 

0.766 

27.1 

688 

0.905 

23.0 

584 

0.770 

27.2 

691 

0.909 

23.1 

587 

0-773 

27-3 

693 

0.912 

23.2 

589 

0.776 

27.4 

696 

0.916 

23-3 

592 

0.779 

27.5 

690 

0.920 

23-4 

594 

0.783 

27.6 

701 

0.923 

23-5 

597 

0.786 

27.7 

704 

0.926 

23.6 

599 

0.789 

27.8 

706 

0.929 

23-7 
23.8 

602 
605 

0.792 
0.796 

3:1 

709 
711 

0.936 

23.9 

607 

0-799 

28.1 

7T4 

o-939 

24.0 

610 

0.803 

28.2 

716 

0.942 

24.1 

612 

0.806 

28.3 

719 

0.946 

294 


GEODETIC  ASTRONOMY. 


§297. 


296.  CORRECTION  TO  MEAN  REFRACTION  AS  GIVEN  IN  §  294. 
DEPENDING  UPON  THE  READING  OF  THE  DETACHED 
THERMOMETER. 

R  =  RM(CB)(CD)(CA).    (See  §  68.) 


d  • 

14 

sg 

c 

il 

Q..J 

as 

CD 

IN 

is 

CD 

d.- 
E£ 

ag 

CD 

H£ 

££ 

Hb 

£3 

Hfe 

<u  a 

£u 

—250 

-3i°.  7 

.172 

20° 

-6°.  7 

.062 

65° 

I8. 

0.972 

110° 

43°.  3 

0.895 

—24 

-31   .1 

.169 

21 

-6  .1 

.060 

66 

18  !9 

0.970 

in 

43   -9 

0.894 

-23 

—30  .6 

.166 

22 

-5   -6 

.058 

67 

19  .4 

o.968 

112 

44   -4 

0.892 

—  22 

—  30  .0 

.164 

23 

-5  -o 

.056 

68 

20  .0 

0.966 

"3 

45   '0 

0.891 

—  21 
—  2O 

-29  .4 

-28  .9 

.161 
.158 

24 
25 

-4  .4 
-3   -9 

•054 
.051 

69 
70 

20    .6 
21    .1 

0.964 
0.962 

114 
"5 

45  -6 
46  .1 

0.890 

0.888 

—  ig 

—28  .3 

.156 

26 

-3  .3 

•°49 

7i 

21     .7 

0.961 

116 

46^.7 

0.886 

—  18 

-27  .8 

•153 

27 

—  2    .8 

.047 

72 

22    .2 

0.959 

117 

47   -2 

0.885 

-17 

-27     .2 

.151 

28 

—  2    .2 

•045 

73 

22    .8 

0.957 

118 

47  -8 

0.884 

-16 

-26  .7 

.148 

29 

—  I    .7 

•043 

74 

23   -3 

0-955 

119 

48  .3 

0.882 

-15 

—  26  .1 

•*45 

30 

—  I    .1 

.041 

75 

23  .9 

0-953 

120 

48  .9 

o.88j 

—  14 

—25  .6 

•M3 

31 

-o  .6 

•039 

76 

24  .4 

0.952 

121 

49   -4 

0.880 

—13 

—  25  .0 

140 

32 

0    .0 

.036 

77 

«5   -o 

0.950 

122 

50  .0 

0.878 

—  12 

—24  .4 

.138 

33 

-fo  .6 

•034 

78 

25   -6 

0.948 

I23 

50  .6 

0.877 

—  II 

-23  .9 

•135 

34 

i   .1 

.032 

79 

26  .1 

0.946 

124 

51   .1 

0.876 

—  10 

-23  -3 

•133 

35 

1    -7 

.030 

80 

26  .7 

0-945 

125 

51   -7 

0.874 

-  9 

—  22    .8 

.130 

36 

2     .2 

.028 

81 

27    .2 

0-943 

126 

52    -2 

0.873 

-  8 

—  22     .2 

.128 

37 

2    .8 

.026 

82 

27   .8 

0.941 

127 

52   .8 

0.871 

—  7 

-21     .7 

•125 

38 

3  -3 

.024 

83 

28  .3 

0-939 

128 

53  -3 

0.870 

-  6 

—  21     .1 

.123 

39 

3  -9 

.022 

84 

28  .9 

0.938 

129 

53  -9 

0.868 

—  5 

—  20  .6 

.  I2O 

40 

4  -4 

.020 

85 

29   -4 

0.936 

130 

54   -4 

0.867 

—  4 
-  3 

—    2 

—  20    .0 
-19    -4 

-18   .9 

.118 

•"3 

42 
43 

5   -o 
5  -6 
6  .1 

.018 
.Ol6 
.014 

86 
87 
88 

30   .0 
30  .6 

0-934 
0-933 
0.931 

297.  CORRECTION  TO 
MEAN  REFRACTION 

—    ! 

-18  .3 

.in 

44 

6  .7 

.OI2 

89 

31    -7 

0.929 

OF    §  2Q4,    DEPEND- 

0 

-17  .8 

.108 

45 

7   -2 

.010 

90 

32     -2 

0.928 

ING  UPON  ATTACHED 

+    I 

-17   .2 

.106 

46 

7  -8 

.008 

32   -8 

0.926 

THERMOMETER. 

2 

3 

-16.7 
—  16  .1 

.103 

.101 

47 
48 

8  -3 
8  -9 

006 
.004 

92 
93 

33   -3 
33  -9 

0.924 
0.923 

R  =  (RM)(CB)(CD)(CA). 

4 

-15  .6 

.099 

49 

9  -4 

.002 

94 

34   -4 

0.921 

See  §  68. 

1 

7 

-15   .0 
-14   .4 
-13  -9 

.096 
•094 
.092 

50 
52 

10    .0 

io  .6 
ii   .1 

I.OOO 

0.998 
0.996 

95 
96 
97 

35  -o 

II:6, 

0.919 
0.917 
0.916 

Temp. 
Fahr. 

Temp. 
Cent. 

CA 

8 
9 

10 

-12   .8 

—  12     .2 

.089 
.087 
.085 

53 
54 
55 

n  .7 

12     .2 

12     .8 

0.994 
0.992 
0.990 

98 
99 

100 

36  .7 
37  -2 
37  -8 

0.914 
0.912 
0.911 

-y? 

—20 

-34°.  4 
-28   .9 

.007 
.006 

—  10 

-23  .3 

.005 

ii 

-II     .7 

.082 

56 

*3  -3 

0.988 

101 

38   .3 

0.909 

O 

-17  .8 

.005 

12 

—  II     .1 

.080 

57 

13  -9 

0.986 

102 

38   -9 

0.908 

4-to 

—  12    .2 

.004 

J3 

—  io  .6 

.078 

58 

14  .4 

0.985 

I03 

39  -4 

0.906 

20 

-  6  .7 

.003 

14 

—  IO    .O 

.076 

59 

15   -o 

0.983 

104 

40  .0 

0.905 

3° 

—  i.i 

.002 

11 

-  9   .4 
-  8  .9 

.073 
.071 

61 

15  -6 
16  .1 

0.981 
0.979 

a 

40  .6 
41   .1 

0.903 
0.902 

40 

£ 

-f-  4    -4 
10  .0 
15  -6 

.001 

.000 

0.999 

17 

-   8   .3 

.069 

62 

16  .7 

0.977 

107 

41    .  7 

0.900 

7° 

21     .1 

0.998 

18 

-  7  -8 

.067 

63 

17    .2 

0.975 

108 

42    .2 

0.899 

80 

26     .7 

0.997 

X9 

-  7   .2 

.064 

64 

17  .8 

0.973 

109 

42  .8 

0.897 

90 

32     .2 

o  996 

100 

37  -8 

0.996 

no 

43   -3 

0.995 

1  20 

48   .9 

0.994 

130 

54   -4 

0.993 

§  298.  TABLES.  295 

298.  DIP  OF  THE  SEA  HORIZON. 


(See  §  78.) 

Height 
of  the 
Eye. 
Feet. 

Dip. 

Height 
of  the 
Eye. 
Feet. 

Dip. 

Height 
of  the 
Eye. 
Feet. 

Dip. 

O 

o' 

oo" 

25 

4' 

54" 

50 

6' 

56" 

I 

o 

59 

26 

5 

oo 

51 

7 

oo 

2 

I 

23 

27 

5 

06 

52 

7 

04 

3 

I 

42 

28 

5 

ii 

53 

7 

08 

4 

I 

58 

29 

5 

17 

54 

7 

12 

5 

2 

ii 

30 

5 

22 

55 

7 

16 

6 

2 

24 

31 

5 

27 

56 

7 

20 

7 

2 

36 

32 

5 

33 

57 

7 

24 

8 

2 

46 

33 

5 

38 

58 

7 

28 

9 

2 

56 

34 

5 

43 

59 

7 

31 

10 

3 

06 

35 

5 

48 

60 

7 

35 

ii 

3 

15 

36 

5 

53 

65 

7 

54 

12 

3 

24 

37 

5 

58 

70 

8 

12 

13 

3 

32 

38 

6 

03 

75 

8 

29 

14 

3 

40 

39 

6 

07 

80 

8 

46 

15 

3 

48 

40 

6 

12 

85 

9 

02 

16 

3 

55 

41 

6 

17 

90 

9 

18 

17 

4 

02 

42 

6 

21 

95 

9 

33 

18 

4 

09 

43 

6 

25 

100 

9 

48 

19 

4 

16 

44 

6 

30 

20 

4 

23 

45 

6 

34 

21 

4 

29 

46 

6 

39 

22 

4 

36 

47 

6 

43 

23 

4 

42 

48 

6 

47 

24 

4 

48 

49 

6 

52 

296 


GEODETIC  ASTRONOMY. 


§299. 


299.      FACTORS    FOR   THE    REDUCTION    OF   TRANSIT   TIME 
OBSERVATIONS. 

The  sign  of  A  is  -j-  except  for  stars  between  the  zenith  and  the  pole,  for  B  it  is  +  except 
for  sub-polars,  and  for  C  it  is  -f-  except  for  sub-polars  with  lamp  WEST,  and  -  except  for  sub- 
polars  with  lamp  EAST.  (See  §§  95,  97,  98.) 

This  top  argument  is  the  star's  declination  ±  S. 


0° 

ioe 

.5° 

20° 

22° 

24° 

26° 

28° 

30° 

32° 

34° 

36° 

£ 

.02 

.02 

.02 

.02 

.02 

.02 

.02 

.02 

.02 

.02 

.02 

.02 

89° 

2 

3 

.04 

.05 

.04 
.05 

.04 

.04 
.06 

.'06 

.04 
.06 

.04 

.06 

.04 
.06 

.04 
.06 

.04 
.06 

a 

.04 

.06 

88 
87 

4 

.07 

.07 

.07 

.07 

!o8 

.08 

.08 

.08 

.08 

.08 

.08 

.09 

86 

5 

.09 

.09 

.09 

.09 

.09 

.10 

.10 

.10 

.10 

.10 

.10 

.11 

85 

6 

.11 

.11 

.11 

.11 

.11 

.11 

.12 

.12 

.12 

.12 

•13 

•13 

83- 

I 

.14 

14 

•14 

•15 

•15 

•'5 

.16 

.16 

.16 

.16 

•17 

82 

g 

.16 

.'16 

.16 

.1.7 

•17 

•  17 

•17 

.18 

.18 

.18 

.19 

.19 

81 

10 

•17 

.18 

.18 

.19 

.19 

.19 

.19 

.20 

.20 

.21 

.21 

.21 

80 

12 

.19 

.21 

.21 

.22 

.22 

.22 

•23 

•23 

'*} 

•24 

•25 

•25 

.24 
.26 
.28 

79 

14 

.24 

•  23 
•  25 

•25 

.26 

,26 

.27 

•27 

•27 

.28 

.29 

.29 

•30 

77 

76 

15 

.26 

.26 

.27 

.28 

.28 

.28 

.29 

.29 

•30 

•31 

•32 

75 

16 

.28 

.28 

•29 

.29 

•3° 

•30 

•3* 

•  3' 

•  32 

•33 

•33 

•34 

74 

17 

.29 

•  3° 

•30 

.31 

.31 

•  32 

•33 

•  33 

•34 

•34 

•35 

•36 

73 

18 

•  31 

•  31 

•32 

•33 

•33 

•34 

•34 

•35 

•36 

.36 

•37 

•38 

72 

19 

•33 

•  33 

•34 

•35 

•35 

•36 

.36 

•  37 

•38 

•38 

•39 

.40 

7» 

20 

•  34 

•  35 

•35 

.36 

•37 

•37 

•38 

•39 

.40 

.40 

.41 

.42 

70 

21 

.36 

.36 

•37 

•38 

•39 

•39 

.40 

.41 

.41 

•  42 

•43 

•44 

69 

22 

•37 

.38 

•39 

.40 

.40 

.41 

•42 

.42 

•43 

•  44 

•45 

.46 

68 

23 

•  39 

.40 

.41 

.42 

•  42 

•43 

•  44 

•  44 

•45 

.46 

•47 

.48 

67 

24 

•  4i 

•42 

•43 

•44 

•45 

•45 

.46 

•47 

.48 

•49 

•50 

66 

25 

•  42 

•  43 

•44 

•45 

.46 

.46 

•  47 

.48 

•49 

•  5° 

•51 

•52 

65 

26 

•  44 

.45 

•45 

•47 

•  47 

.48 

•49 

•5° 

•51 

•  S2 

•53 

•54 

64 

27 

•  45 

46 

•47 

.48 

•49 

•5° 

.51 

•52 

•54 

•55 

.56 

63 

28 

•  47 

.48 

•49 

•5° 

•51 

•51 

•52 

•53 

•54 

•55 

•57 

.58 

62 

29 

.48 

•  49 

•50 

•52 

•  52 

•53 

•54 

•55 

•56 

•57 

•  58 

.60 

61 

3° 

•  50 

•52 

•53 

•54 

•55 

•56 

•57 

•58 

•59 

.60 

.62 

60 

31 

•S2 

.52 

•53 

•55 

•56 

.56 

•57 

•58 

•59 

.61 

'  .62 

.64 

59 

32 

•53 

•54 

•56 

•57 

.58 

•59 

.60 

.61 

•63 

.64 

58 

33 
34 
35 

•  57 

•55 

•58 
•59 

.58 
•59 
.61 

:g 

.62 

.60 
.61 
•63 

.61 

.62 
.64 

.62 

'§ 

•65 

•63 

:§ 

.64 
.66 

.68 

.66 
.67 
.69 

'67 
.69 

11 

55 

\ 

36 

•59 

.60 

.61 

•63 

•63 

.64 

•65 

.67 

.68 

.69 

.71 

•73 

54 

37 

.60 

.61 

.62 

.64 

.65 

.66 

.67 

.68 

.70 

•73 

•74 

53 

38 

.62 

•63 

.64 

.66 

.66 

.67 

.69 

.70 

•73 

•74 

.76 

52 

39 

•63 

•64 

•65 

.67 

.68 

.69 

.70 

•73 

.76 

.78 

40 

•64 

•65 

.67 

.68 

.69 

.70 

.72 

•73 

•74 

.76 

•77 

•79 

50 

4» 

.66 

.67 

.68 

.70 

•71 

•72 

•73 

•74 

.76 

•77 

•79 

.81 

49 

42 

.67 

.68 

.69 

•72 

•73 

•74 

.76 

•77 

•79 

.81 

•83 

48 

43 

.68 

.69 

•  71 

•73 

•74 

.76 

•77 

•79 

.80 

.82 

.84 

47 

44 

.69 

•71 

•  72 

•74 

•75 

.76 

•77 

•79 

.80 

.82 

.84 

.86 

46 

45 

•71 

.72 

•73 

•75 

.76 

•77 

•79 

.80 

.82 

•83 

•85 

.87 

45 

299. 


TABLES. 


297 


FACTORS   FOR   THE   REDUCTION   OF   TRANSIT  TIME 
OBSERVATIONS. 

This  top  argument  is  the  star's  declination  ±  S. 


o° 

30° 

32° 

34° 

36° 

46° 

•72 

•73 

•77 

.78 

•79 

.80 

.82 

•83 

•85 

87 

.89 

44° 

•73 

.76 

.78 

•79 

.80 

.81 

.83 

.84 

.86 

.88 

.90 

43 

48 

•74 

.76 

•77 

•79 

.80 

.81 

•  83 

.84 

.86 

.88 

.90 

.92 

42 

49 

•75 

•77 

.78 

.80 

.81 

•83 

.84 

.86 

.87 

.89 

.91 

•93 

41 

50 

•77 

.78 

•79 

.82 

•83 

.84 

•  85 

.87 

.89 

.90 

•92 

•95 

40 

Si 

•78 

•79 

.80 

•83 

.84 

•85 

.87 

.88 

.90 

.92 

•94 

.96 

39 

52 

•79 

.80 

.82 

.84 

•85 

.86 

.88 

.89 

•93 

•95 

•97 

38 

53 

.80 

.81 

•83 

•Si 

.86 

•  87 

.89 

.91 

•  92 

•94 

.96 

•99 

37 

54 

.81 

.82 

.84 

.86 

.87 

.89 

.90 

.92 

•93 

•95 

.98 

I.OO 

36 

55 

.82 

•83 

•85 

•87 

.88 

.90 

.91 

•93 

•95 

•97 

•99 

I.OI 

35 

56 

•83 

.84 

.86 

.88 

.89 

.91 

•92 

•94 

.96 

•98 

I.OO 

1.02 

34 

57 

.84 

•85 

.87 

.89 

.90 

.92 

•93 

•95 

•97 

•99 

I.OI 

1.04 

33 

58 

•85 

.86 

.88 

.90 

.91 

•93 

•94 

.96 

.98 

I.OO 

1.  02 

1.05 

32 

59 

.86 

•87 

.89 

.91 

•  92 

•94 

•95 

•97 

•99 

I.OI 

1.03 

I.  06 

31 

6o 

.87 

.88 

.90 

.92 

•93 

•95 

.96 

.98 

I.OO 

i  .02 

1.04 

1.07 

30 

61 

•87 

.89 

.91 

•93 

•94 

.96 

•97 

•99 

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1.03 

1.  06 

I.  08 

29 

62 

.88 

.90 

.91 

•94 

•95 

•97 

.98 

I.OO 

.02 

1.04 

1.  06 

1.09 

28 

63 
64 

.89 
.90 

.91 
•91 

.92 
•93 

9 

.96 
•97 

.98 
.98 

•99 
.00 

I.OI 
1.02 

.04 

1:3 

1.07 
1.  08 

I.  10 

I  .  II 

27 
26 

65 

.91 

.92 

•94 

.96 

.98 

•99 

.01 

1.03 

.05 

1.07 

1.09 

1.  12 

25 

66 

.91 

•93 

•95 

•97 

•99 

I.OO 

.02 

1.04 

.06 

1.  08 

1  .10 

I.13 

24 

67 

.92 

•94 

•95 

.98 

•99 

I.OI 

.02 

1.04 

.06 

1.09 

I.  II 

I.I4 

23 

68 

•93 

•94 

.96 

•99 

I.OO 

1.02 

.03 

1.05 

.07 

1.09 

1.  12 

I.I5 

22 

69 

•93 

•95 

•97 

•99 

I.OI 

I.  O2 

.04 

1.  06 

.08 

I.  10 

1  .  13 

I  •  15 

21 

70 

•94 

•95 

•97 

I.OO 

I.OI 

1.03 

•05 

1.  06 

.09 

I.  II 

«-«3 

1.16 

20 

71 

•95 

.96 

•98 

1.01 

1.02 

1.04 

.05 

1.07 

.09 

1.  12 

1.14 

1.17 

19 

72 

•95 

•97 

.98 

I.OI 

1.03 

1.04 

.06 

I.  08 

.10 

I.  12 

1-15 

1.17 

18 

73 

.96 

•97 

•99 

1.02 

1.03 

I.OS 

.06 

1.  08 

.10 

1-13 

1.18 

17 

74 

.96 

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I.OO 

1.02 

1.04 

I.OS 

.07 

1.09 

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1-13 

1.16 

1.19 

16 

75 

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I.OO 

1.03 

1.04 

1.  06 

.08 

1.09 

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1.14 

1.16 

1.19 

15 

76 

•97 

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I.OO 

1.03 

1.05 

1.06 

.08 

I.  10 

.12 

1.14 

1.17 

1.20 

14 

77 

•97 

•99 

I.OI 

1.04 

1.05 

1.07 

.08 

1.  10 

.13 

1.15 

1.17 

I.  2O 

13 

78 

.98 

•99 

I.OI 

1.04 

I.OS 

1.07 

.09 

I.  II 

•13 

1.15 

1.18 

I.  21 

12 

79 

.98 

.00 

1.02 

1.04 

I.  06 

i.  08 

.09 

I.  II 

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1.16 

1.18 

I.  21 

II 

80 

.98 

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1.02 

1.05 

1.06 

i.  08 

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I.  12 

.14 

1.16 

1.19 

1.22 

10 

81 

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1.05 

1.07 

i.  08 

.10 

1.  12 

.14 

1.17 

1.19 

1.22 

9 

82 

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1.03 

1.05 

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1.17 

1.19 

I  .22 

8 

83 

•99 

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1.  06 

1.07 

1.09 

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.15 

1.17 

1.20 

1.23 

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84 

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1.03 

1.  06 

1.07 

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.15 

1.17 

1.  2O 

1.23 

6 

85 

I.OO 

.01 

1.03 

1.  06 

1.07 

1.09 

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»**3 

•15 

1.17 

1.20 

1.23 

5 

86 

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.01 

1.03 

1.06 

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1.18 

1.20 

1.23 

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87 

1  .00 

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1.03 

1.06 

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1.09 

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1.13 

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1.  18 

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1.23 

3 

88 

Qrt 

I.OO 

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1-03 

i.  06 

I  O6 

i.  08 

1.09 

.11 

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1.18 

1.20 

1.23 

2 

°9 
go 

1  .00 

1  .00 

.02 

.02 

1.04 

1.04 

i!o6 

i!o8 

1.09 
1.09 

I.  II 

1.18 

I  .21 

1.24 

0 

The  bottom  line  on  this  page  is  the  collimation  factor  C  (=  sec  5). 


298 


GEODETIC  ASTRONOMY. 


§299. 


FACTORS    FOR   THE    REDUCTION   OF  TRANSIT  TIME 
OBSERVATIONS. 

This  top  argument  is  the  star's  declination  ±  S. 


38° 

40° 

4- 

42° 

43° 

44° 

45° 

46° 

47° 

48° 

49° 

50° 

J. 

.02 

.02 

.02 

.02 

.02 

.02 

.02 

.02 

•03 

•03 

4- 

2 

.04 

•°5 

•05 

•05 

•05 

•05 

•05 

•°5 

•05 

•05 

•05 

•°5 

88 

3 

.07 

.07 

.07 

.07 

.07 

.07 

.07 

.07 

.08 

.08 

.08 

.08 

87 

4 

.09 

.09 

.09 

.09 

.10 

.10 

.10 

.10 

.10 

.10 

.11 

.  ii 

86 

0. 

5 
6 

°5 
g. 

7 

•15 

.16 

.16 

.16 

.18 

.18 

.18 

.19 

.19 

04 
83 

8 

.18 

.18 

.18 

-19 

.19 

.19 

.20 

.20 

.20 

.21 

.21 

.22 

82 
81 

9 

10 

.22 

•23 

•23 

•23 

•24 

.24 

•25 

•25 

•25 

.26 

.26 

•27 

80 

ii 

12 

:3 

•25 
.27 

•25 
.27 

.26 
.28 

.26 
.28 

•27 
•29 

•27 
•29 

.28 
•30 

.28 
•30 

.28 
•31 

•29 
•32 

•30 
•32 

79 
78 

13 

•29 

.29 

.30 

•3° 

•31 

•  3* 

•32 

•32 

•33 

•34 

•34 

•35 

77 

M 

•32 

•32 

•33 

•33 

•34 

•34 

•35 

•35 

•36 

•37 

.38 

76 

IS 

•33 

•34 

•34 

•35 

•35 

•36 

•37 

•37 

•38 

•39 

•39 

.40 

75 

16 

•35 

•36 

•37 

•37 

•38 

•38 

•39 

.40 

.40 

.41 

.42 

•43 

74 

17 

•37 

•38 

•39 

•39 

.40 

.41 

.41 

.42 

•43 

•44 

•45 

•45 

73 

18 

•39 

•4° 

.41 

•42 

.42 

•43 

•44 

•44 

•45 

.46 

•47 

.48 

72 

19 

.41 

•42 

•43 

•44 

•45 

•45 

.46 

•47 

.48 

•49 

•5° 

•51 

20 

•43 

•45 

•45 

.46 

•  47 

.48 

.48 

.49 

•50 

•52 

•53 

70 

21 

•45 

•47 

•47 

.48 

•49 

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•S* 

•52 

•S2 

•54 

•55 

•56 

69 

22 

.48 

•49 

•5° 

•5° 

.51 

•52 

•53 

•54 

•55 

.56 

-.57 

•58 

68 

23 

•5° 

•52 

•53 

•53 

•54 

•55 

.56 

•57 

.60 

.61 

67 

24 

•52 

•53 

•54 

•55 

.56 

•57 

•58 

•59 

.60 

.61 

.62 

•63 

66 

25 

•54 

•55 

.56 

•57 

•58 

•59 

.60 

.61 

.62 

•63 

.64 

.66 

65 

26 

27 

-56 

rR 

•57 

t 

•  59 
61 

.60 
.62 

.61 
.61 

.62 
64. 

•63 
fic 

.64 
67 

•S 

.67 

60 

.68 

64 
61 

28 
29 

.60 
.61 

^63 

.62 
.64 

•63 

•  65 

.64 
.66 

1 

.66 

.69 

.68 
.70 

.69 

.70 
.72 

•72 

•73 
•75 

°3 
62 
61 

30 

•63 

•65 

.66 

•  67 

.68 

.69 

•71 

•  72 

•73 

•75 

.76 

•78 

60 

3* 

65 

.67 

.68 

.69 

.70 

•72 

•73 

•74 

•75 

•77 

•  78 

.80 

59 

32 

33 

.67 

.69 

.69 
•71 

•7° 
•72 

•71 
•73 

•  72 
•74 

:764 

•75 
•77 

.76 

•78 

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•79 
.81 

•83 

.82 
•85 

58 
57 

34 

•71 

•73 

•74 

•75 

.76 

.78 

•79 

.80 

.82 

.84 

•85 

.87 

56 

35 

•73 

•75 

.76 

•77 

.78 

.80 

.81 

•83 

.84 

.86 

•87 

.89 

55 

36 

•75 

•77 

•78 

•79 

.80 

.82 

•83 

.85 

.86 

.88 

.90 

.91 

54 

P 

.76 
•78 

•79 
.80 

.80 
.82 

.81 

83 

.82 
.84 

•a 

3 

•  87 
•89 

.88 
.90 

.90 
.92 

.92 
•94 

•94 
.96 

53 
52 

39 

.80 

.82 

•  83 

.85 

.86 

.87 

.89 

.92 

•94 

.96 

.98 

51 

40 

.82 

.84 

-85 

.86 

.88 

.89 

.91 

•93 

•  94 

.96 

.98 

I.OO 

50 

41 

•83 

.86 

•  87 

.88 

.90 

.91 

•93 

•  94 

.96 

.98 

I.OO 

.02 

49 

42 

.85 

•  87 

.89 

.90 

.91 

•93 

•95 

.96 

.98 

I.OO 

1.02 

.04 

48 

43 

.86 

.89 

.90 

.92 

•93 

•95 

.96 

.98 

I.OO 

i  .02 

1.04 

.06 

47 

44 

.88 

.91 

•92 

•93 

•95 

•  97 

.98 

I.OO 

1.02 

1.04 

1.  06 

.08 

46 

45 

.90 

.92 

•94 

•95 

•97 

.98 

I.OO 

1.02 

1.04 

i.  06 

I.  08 

.IO 

45 

299- 


TABLES. 


299 


FACTORS    FOR   THE    REDUCTION    OF   TRANSIT   TIME 
OBSERVATIONS. 

This  top  argument  is  the  star's  declination  ±  5. 


38° 

40- 

41° 

42° 

43° 

44° 

45° 

46° 

47° 

48° 

49° 

50° 

c 

£ 

46° 

.91 

•94 

•95 

•97 

.98 

i  .00 

1.02 

1.04 

1.05 

1.07 

.JO 

I.  12 

44° 

47 

•93 

•95 

•97 

.98 

I.OO 

1.02 

1.03 

1.05 

1.07 

1.09 

.11 

I.I4 

43 

48 

•94 

•97 

.98 

I.  00 

1.02 

1.03 

1.05 

1.07 

1.09 

i  .11 

•-3 

1.16 

42 

49 
50 

.96 

•97 

•99 

1.  00 

I.  00 

i  .01 

1.02 
1.03 

1.03 
1.05 

J:3 

1.07 
I.  08 

1.09 

l.IO 

i.  ii 

I.  12 

'•'3 
1.14 

•'5 

•'7 

1.17 
1.  19 

40 

51 

•99 

I.OI 

1.03 

1.05 

1.  06 

i.  08 

l.IO 

1.  12 

I.I4 

1.16 

.18 

I.  21 

39 

52 

I.  00 

1.03 

1.04 

I.  06 

1.  08 

l.IO 

I.  II 

'•'3 

'•'5 

1.18 

.20 

1.23 

3? 

53 

1.  01 

1.04 

i.  06 

1.07 

1.  09 

I.  II 

'•'3 

'•'5 

I.T7 

1.19 

.22 

1.24 

37 

54 

ce 

1.03 

I  06 

I  O7 

1.07 
i.  08 

1.09 
I  •  IO 

I.  IX 
I  •  12 

I.  12 

1.14 

1.19 

I  .21 

•23 

1.26 

36 

jj 

56 

57 

:s 

A  .  U/ 

i.  08 
i.  09 

1.  10 

I.  II 

1.  12 

'•'3 

'•'7 

1.19 

1.19 

I.  21 

1.22 
1.23 

1.24 

•25 

.26 
.28 

1.27 

1.29 
I.3I 

35 

34 
33 

58 

i.  08 

i  .11 

I.  12 

1.14 

1.16 

1.18 

1.20 

1.22 

1.24 

1.27 

.29 

1.32 

32 

59 

i.  09 

I.  12 

I.I4 

'•'5 

1.  17 

1.19 

I.  21 

1.23 

1.26 

1.28 

•3' 

'•33 

3' 

60 

1.  10 

'•'3 

'•'5 

1.17 

1.18 

1.20 

1.22 

'•25 

1.27 

1.29 

•32 

'•35 

30 

61 

62 

I.  II 

1.14 

1.16 

1.18 

1.20 

1.22 

1.24 

1.26 

1.28 

'•3' 

•33 

1-36 

29 

63 

'•'3 

1.16 

1.18 

1.20 

1.22 

1.24 

1.25 
1.26 

1.27 
1.28 

1.29 

1.32 
'•33 

•35 
•36 

'•37 
'•39 

28 
27 

64 

1.14 

1.17 

1.19 

I.  21 

'•23 

'•25 

1.27 

1.29 

1-32 

'•34 

•37 

1.40 

26 

65 

'•'5 

1.18 

1.20 

1.22 

1.24 

1.26 

1.28 

I.30 

'•33 

'•35 

.38 

1.41 

25 

66 

1.  16 

1.19 

I.  21 

1.23 

'•25 

1.27 

1.29 

I.32 

'•34 

'•37 

39 

1.42 

24 

67 

1.17 

1.20 

1.22 

1.24 

1.26 

1.28 

'•33 

'•35 

1.38 

40 

23 

68 

1.18 

I.  21 

1.23 

'•25 

1.27 

1.29 

'•3' 

'•33 

1.36 

'•39 

•4' 

1.44 

22 

69 

x.i8 

I  .22 

1.24 

I  .26 

1.28 

1.30 

1.32 

'•34 

i  .37 

1.40 

.42 

1.45 

21 

70 

1.19 

'•23 

1.25 

1.26 

1.28 

'•33 

'•35 

1.38 

1.40 

43 

1.46 

2O 

1.20 

1.23 

'•25 

1.27 

1.29 

x.3t 

'•34 

1.36 

'•39 

1.41 

44 

1.47 

'9 

72 

1  .21 

1.24 

1.26 

1.28 

I.30 

1-32 

i'34 

'•39 

1.42 

45 

A.6 

1.48 

18 

TT 

73 
74 

1.22 

I.  2J 

1.27 

1.29 

I.3I 

'•34 

1.36 

1.38 

1.41 

'•43 
'•44 

c 
46 

i  .49 
'•49 

*/ 

16 

75 

'•23 

1.26 

1.28 

1.30 

1.32 

'•34 

'•37 

'•39 

1.42 

1.44 

47 

1.50 

15 

76 

1.23 

1.27 

1.29 

I.3I 

'•33 

'•35 

'•37 

1.40 

1.42 

'•45 

48 

'•5' 

14 

77 

1.24 

1.27 

1.29 

'•3' 

'•33 

'•35 

1.38 

1.40 

'•43 

1.46 

48 

1.52 

78 

1.24 

1.28 

1.30 

1.32 

1.36 

1.38 

1.41 

'•43 

1.46 

49 

1.52 

12 

79 

1.25 

1.28 

1.30 

1.32 

'•34 

1.36 

'•39 

1.41 

'•44 

1.47 

50 

'•53 

II 

80 

'•25 

1.29 

1.30 

'•33 

'•39 

1.42 

1.44 

'•47 

5° 

'•53 

IO 

81 

'•25 

1.29 

I.3I 

'•33 

'•35 

'•37 

1.40 

1.42 

'•45 

1.48 

5' 

1-54 

9 

82 

1.26 

1.29 

'•3' 

'•33 

'•35 

1.38 

1.40 

1.43 

1.48 

51 

1.54 

8 

83 

1.26 

1-30 

1.32 

'•34 

1.36 

1.38 

1.40 

'•43 

\'446 

1-48 

5' 

1-54 

7 

84 

1.26 

1.30 

1.32 

1.36 

1.38 

1.41 

'•43 

i.46 

1.49 

52 

'•55 

6 

85 

X.26 

I.TO 

'•32 

'•34 

1.36 

1.38 

1.41 

'•43 

i.46 

1.49 

52 

'•55 

5 

86 

1.27 

1.30 

1.32 

'•34 

1.36 

'•39 

1.41 

1.44 

1.46 

1.49 

S2 

'•55 

4 

87 

1.27 

1-30 

'•32 

'•39 

1.41 

1.44 

1.46 

1-49 

52 

'•55 

3 

88 
89 

1.27 
1.27 

1.30 

'•3' 

1.32 
'•33 

'•35 

'•37 

J  -39 

1.41 
1.41 

1.44 
'•44 

i.46 
1.47 

1.49 
1.49 

52 
52 

::p 

2 
I 

90 

1.27 

'•37 

'•39 

1.41 

'•44 

'•47 

1.49 

52 

1.56 

O 

The  bottom  line  of  this  page  is  the  collimation  factor  C  (=  sec  5). 


300 


GEODETIC  ASTRONOMY. 


§299. 


FACTORS    FOR   THE    REDUCTION   OF  TRANSIT  TIME 
OBSERVATIONS. 

This  top  argument  is  the  star's  declination  ±  5. 


5.° 

* 

53° 

54° 

55° 

56- 

57° 

58° 

59° 

60° 

<* 

61° 

1° 

2 

!o6 

3 

.03 

a 

.03 

•°3 
.06 

.03 

.07 

.03 

.03 
.07 

.04 
.07 

.04 
.07 

1° 

3 

.08 

.08 

.09 

.09 

.09 

.09 

.10 

.10 

.10 

.10 

.11 

.11 

87 

4 

.11 

.  ii 

.  12 

.12 

.12 

.12 

:3 

•T| 

.14 

•14 

.14 

.14 
.  18 

86 

0- 

5 
6 

•  17 

•  17 

•J7 

.18 

.18 

.19 

.19 

.20 

.20 

.21 

.21 

.22 

°5 
84 

7 
8 

.19 

.22 

.20 
•23 

.20 

•23 

.21 

.24 

.21 

.24 

.22 
•25 

.22 

.26 

a 

.24 
.27 

a 

:3 

•25 

•29 

83 
82 

9 

•25 

•25 

.26 

.26 

.27 

.28 

.29 

•29 

•30 

.31 

.32 

•32 

81 

10 

.28 

.28 

.29 

•30 

•30 

•31 

•32 

•33 

•34 

•35 

•35 

•36 

80 

ii 

•30 

•31 

•32 

•32 

•33 

•34 

•35 

•36 

•37 

•38 

•39 

•39 

79 

12 

•35 

•35 

•  36 

•37 

•38 

•39 

.40 

•42 

.42 

•43 

78 

•2*7 

.08 

on 

•  4O 

4* 

•  4-2 

•  44 

46 

1  J 

M 

.38 

39 

.40 

.41 

.42 

•43 

•44 

.46 

•47 

.48 

•49 

•So 

77 

76 

15 

.41 

•42 

•43 

•44 

•45 

.46 

.48 

•  49 

•50 

•52 

•53 

•53 

75 

16 

•44 

•45 

.46 

•47 

.48 

•49 

.51 

•52 

•54 

•55 

•56 

•57 

74 

18 

.46 

•49 

•47 
•50 

•49 

•So 
•53 

•5* 
•54 

•52 

•55 

•54 
•57 

9 

:g 

•58 

.62 

•59 

.60 

.64 

73 
72 

19 

20 

•52 
•54 

•53 
•56 

•54 
•57 

1 

.60 
•63 

.61 
.64 

:§ 

;66I 

\i 

.67 
.70 

70 

21 

•57 

•58 

•59 

.61 

.62 

.64 

.66 

.68 

.70 

•72 

•73 

•74 

69 

22 

.60 

.61 

.62 

.64 

•65 

.67 

.69 

•  71 

•73 

•75 

.76 

•77 

68 

23 

.62 

•63 

•65 

.66 

.68 

.70 

.72 

•74 

.76 

•78 

•79 

.81 

67 

24 
25 

:657 

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.69 

.68 
.70 

.69 
.72 

•71 
•74 

•73 
.76 

•75 
•78 

•77 
.80 

•79 

.82 

.81 
•85 

11 

.84 
.87 

66 
65 

26 

.70 

•  71 

•73 

•75 

•  76 

.78 

.80 

•83 

•85 

.88' 

.89 

.90 

64 

27 

•  72 

•74 

•75 

•77 

•79 

.81 

•83 

.86 

.88 

.91 

.92 

•94 

63 

28 

•75 

.76 

•78 

.80 

.82 

.84 

.86 

.89 

.91 

•94 

•95 

•97 

62 

29 

•77 

•79 

.81 

.82 

.84 

•87 

.89 

.91 

•94 

•97 

.98 

.00 

61 

30 

•79 

.81 

.83 

•85 

•87 

.89 

.92 

•94 

•97 

1.  00 

i  .01 

.03 

60 

31 

.82 

.84 

.86 

.88 

.90 

.92 

•95 

•97 

1.  00 

•03 

•05 

.06 

59 

32 

.84 

.86 

.88 

.90 

.92 

•95 

•97 

I.  00 

1.03 

.06 

.08 

.09 

58 

33 

•87 

.88 

.91 

•93 

•95 

•97 

1.  00 

1.03 

i.  06 

.09 

.11 

.12 

57 

34 

.89 

91 

•93 

•95 

•97 

.00 

1.03 

1.05 

1.09 

.12 

•'4 

•J5 

56 

35 

.91 

•93 

•95 

.98 

I.  00 

•°3 

1.05 

i.  08 

i.  ii 

•15 

.16 

.18 

55 

36 

•93 

•95 

.98 

.00 

•03 

•05 

i.  08 

i.  ii 

1.14 

.18 

.19 

.21 

54 

37 

.96 

.98 

.00 

.02 

.05 

.08 

1.  10 

1.14 

1.17 

.20 

.22 

•24 

53 

38 

.98 

.00 

.02 

•05 

.07 

.10 

I>13 

1.16 

i  .20 

•23 

•25 

•27 

52 

39 

.00 

.02 

•05 

•0? 
no 

.10 

.12 

I'1| 

1.19 

1.22 

.26 

.28 

•30 

51 

40 

41 
42 

.04 
.06 

.04 

.07 
.09 

.07 

.09 
.11 

»uy 
.  12 

•M 

•17 
.20 

1.20 
1-23 

\\ll 

1.27 
1.30 

•3* 

•33 
•36 

•33 

1 

5° 

49 

48 

43 

.08 

.11 

•T3 

.16 

.19 

.22 

1-25 

1.29 

1.32 

•  36 

•39 

47 

44 

.10 

•13 

•IS 

.18 

.21 

.24 

1.28 

i-35 

•39 

•43 

46 

45 

.  12 

.20 

•23 

.26 

1.30 

i'33 

'•37 

.41 

•44 

•46 

45 

§299- 


TABLES. 


301 


FACTORS   FOR   THE   REDUCTION    OF   TRANSIT   TIME 
OBSERVATIONS. 

This  top  argument  is  for  the  star's  declination  ±  6. 


5-' 

52° 

53° 

54° 

55° 

56° 

57° 

58° 

59° 

60° 

*P 

61° 

4«° 

.14 

j>i7 

.19 

1.22 

1-25 

1.29 

•32 

1.36 

1.40 

•44 

I.46 

.48 

44° 

8 

.16 
.18 

.19 

.21 

.21 
•23 

1:3 

1.27 

1.30 

*-33 

$ 

1.38 

1.40 

1.42 
'•44 

3 

1.49 
1.50 

•53 

43 
42 

49 

.20 

•23 

•25 

1.28 

1.32 

•39 

1.42 

1.47 

•51 

"•S3 

•56 

50 

.22 

•24 

•27 

1.30 

i-34 

1-37 

.41 

1.44 

1.49 

•53 

1.56 

-58 

40 

5I 

•23 

.26 

•29 

1.32 

!.35 

i-39 

•43 

i.47 

i.S« 

•55 

'•58 

.60 

39 

52 

•25 

.28 

•  3' 

i  •  34 

i-37 

1.41 

•45 

1.49 

1-53 

•58 

i.  60 

•  63 

38 

53 

.27 

•3° 

•33 

i  36 

J-39 

•47 

1-55 

.60 

1.62 

.65 

37 

54 
55 

.29 

•3° 

•33 

1.38 

1.41 

I'll 

•49 
•50 

1-53 
1-55 

i-57 

1-59 

.62 

.64 

1.64 
1.66 

•67 
.69 

36 
35 

56 

11 

59 

•3* 
•33 

•35 

•39 

•38 
•39 
.41 
.42 

1.41 

'•44 
1.46 

1-45 
1.46 
1.48 
1.49 

1.48 
1.50 
1.52 
i-53 

•52 

31 

•57 

I'A 

i.  60 
1.62 

i.6r 
1.63 
1.65 
1.66 

.66 
.68 
.70 

1.68 
1.70 
1.72 

•73 
•75 
•77 

34 
33 
32 
3* 

60 

[38 

.41 

•44 

1-47 

1.51 

I-55 

•59 

i.63 

1.68 

•73 

1.76 

•79 

30 

61 

•39 

•42 

•45 

1.49 

1-53 

1.56 

.61 

i.6S 

1.70 

•75 

1.78 

.80 

29 

62 
63 

.40 

•42 

•43 

•45 

:!i 

1.50 

i-54 
!'55 

1.58 
i-59 

.62 
.64 

1.67 

1.68 

1.71 
1-73 

•77 
•78 

1.79 
t.8i 

.82 
.84 

28 
27 

64 

•43 

.46 

•49 

1  -53 

1.61 

•65 

1.70 

i.7S 

.80 

1.83 

•85 

26 

65 

•  44 

•47 

•51 

1-54 

xip 

1.62 

.66 

1.71 

1.76 

.81 

1.84 

.87 

25 

66 

•45 

.48 

•52 

1-55 

1-59 

1.63 

.68 

1.72 

1.77 

•83 

1.85 

.88 

24 

67 

.46 

•5° 

•53 

i.  60 

1.65 

.69 

i-74 

1.79 

.84 

1.87 

.90 

23 

68 

•47 

•51 

•54 

lip 

1.62 

1.66 

.70 

i.  80 

1.88 

22 

69 

.48 

•52 

•55 

1-59 

1.63 

1.67 

•71 

1.76 

x.8i 

.87 

1.90 

•93 

21 

70 

.49 

•53 

.56 

i.  60 

1.64 

1.68 

•73 

1.77 

1.82 

.88 

1.91 

•94 

2O 

72 

•5° 
•51 

•54 
•54 

•57 
•58 

1.61 
1.62 

1.65 
1.66 

1.69 
1.70 

•74 
•75 

1.78 
i.  So 

1.84 
1-85 

.89 
.90 

1.92 

•95 
.96 

18 

73 

•52 

•55 

•59 

1.63 

1.67 

1.71 

.76 

i.  80 

1.86 

.91 

1.94 

•97 

17 

74 

•53 

•56 

.60 

1.63 

1.68 

1.72 

.76 

1.81 

1.87 

•92 

i-95 

.98 

16 

75 

•53 

•57 

.60 

1.64 

1.68 

i-73 

•77 

1.82 

1.88 

•93 

1.96 

•99 

15 

76 

•54 

.58 

.61 

1.65 

1.69 

1-73 

.78 

1.83 

1.88 

•94 

1.97 

.00 

M 

77 

•55 

•58 

.62 

i  .•66 

1.70 

1.74 

•79 

1.84 

1.89 

•95 

1.98 

.01 

13 

78 

•55 

•59 

.62 

1.66 

1.70 

1.75 

.80 

1.85 

1.90 

.96 

1.99 

.02 

12 

79 

•56 

•59 

•63 

1.67 

1.71 

1.76 

.80 

1.85 

1.91 

.96 

1.99 

.02 

IT 

80 

.56 

.60 

.64 

1.67 

1.72 

1.76 

.81 

1.86 

1.91 

•97 

2.  GO 

•°3 

IO 

81 

•57 

.60 

.64 

1.68 

1.72 

1.77 

.81 

1.86 

1.92 

.98 

2.01 

04 

9 

82 

•57 

.61 

.64 

T.68 

*-73 

1.77 

.82 

1.87 

1.92 

.98 

2.OI 

.04 

8 

83 

-58 

.61 

•65 

1.69 

1.77 

.82 

1.87 

i-93 

•99 

2.02 

•05 

7 

84 

•58 

.62 

•65 

1.69 

J-73 

1.78 

•83 

1.88 

•99 

2.  O2 

.05 

6 

85 

•58 

.62 

•65 

1.69 

1-74 

1.78 

.83 

1.88 

1-93 

•99 

2.02 

•05 

5 

86 

•59 

.62 

.66 

1.70 

1.74 

1.78 

•83 

1.88 

1.94 

2.00 

2.03 

.06 

4 

87 

•59 

.62 

.66 

1.70 

1.74 

1.79 

•83 

1.88 

1.94 

2.00 

2.03 

.06 

3 

88 

•59 

.62 

.66 

1.70 

1.74 

1.79 

•83 

1.89 

1.94 

2.OO 

2.03 

.06 

2 

89 

•59 

.62 

.66 

1.70 

1.74 

1.79 

.84 

1.89 

1.94 

2.00 

2.03 

.06 

I 

90 

•59 

.62 

.66 

1.70 

1.74 

1.79 

.84 

1.89 

1.94 

2.OO 

2.03 

.06 

O 

The  bottom  line  on  this  page  is  the  collimation  factor  C  (=  sec  8). 


302 


GEODETIC  ASTRONOMY. 


299. 


FACTORS    FOR   THE    REDUCTION    OF  TRANSIT  TIME 
OBSERVATIONS. 

This  top  argument  is  the  star's  declination  ±  5. 


*P 

62° 

62*° 

63° 

63*° 

64° 

64*° 

65° 

65i° 

66° 

66*° 

67° 

(. 

.04 

.04 

.04 

.04 

.04 

.04 

.04 

.04 

.04 

.04 

.04 

.04 

89° 

2 

.07 

.07 

.08 

.08 

.08 

.08 

.08 

.08 

.08 

.09 

.09 

.09 

88 

3 

.  ii 

.  ii 

.11 

.  12 

.  12 

.12 

.12 

.12 

•13 

.13 

87 
86 

5 

.18 

.19 

.19 

.19 

.19 

.20 

.20 

.21 

.21 

.21 

.22 

.22 

85 

6 

.22 

.22 

•23 

•23 

•23 

.24 

•24 

•25 

•25 

.26 

.26 

.27 

84 

7 

.26 

.26 

.26 

•27 

.27 

.28 

.28 

•29 

•29 

•30 

•31 

•31 

83 

8 

.29 

•30 

•30 

•31 

•31 

•32 

•32 

•33 

•34 

•34 

•35 

•36 

82 

9 

•33 

•33 

•34 

•35 

•35 

•36 

•36 

•37 

•38 

•39 

•39 

.40 

81 

10 

•36 

•37 

•38 

•38 

•39 

.40 

.40 

.41 

.42 

•43 

•43 

•44 

80 

ii 

.40 

.41 

.41 

•42 

•43 

•44 

•44 

•45 

.46 

•47 

.48 

•49 

79 

12 

•44 
•47 

3 

•45 
•49 

.46 
•50 

•47 
•5° 

•47 

.48 
•52 

•49 
•53 

•50 
•54 

•55 

'It 

•53 
•58 

78 
77 

14 

•5* 

•  52 

•53 

•54 

•55 

•56 

•58 

•59 

.61 

.62 

76 

15 

•54 

•55 

•56 

•57 

•58 

•59 

.60 

.61 

.62 

.64 

•65 

.66 

75 

i6 

•58 

•59 

.60 

.6! 

.62 

.63 

.64 

•65 

.66 

.68 

.69 

•71 

74 

17 

.61 

.62 

•63 

.64 

.66 

.67 

.68 

.69 

.70 

•  72 

•73 

•75 

73 

18 

•65 

.66 

•67 

.68 

.69 

•7o 

•72 

•73 

•74 

.76 

•77 

•79 

72 

ig 

.68 

.69 

.70 

•72 

•73 

•74 

.76 

•77 

.78 

.80 

.82 

•83 

71 

20 

.72 

•73 

•74 

•75 

•77 

•79 

•79 

.81 

•83 

.84 

.86 

.88 

70 

21 

•75 

.76 

•78 

•79 

.80 

.82 

•83 

•85 

.86 

.88 

.90 

•92 

69 

22 

.78 

.80 

.81 

.82 

.84 

•85 

•87 

.89 

.90 

•  92 

•94 

.96 

68 

23 

.82 

•83 

•85 

.86 

.88 

.89 

.91 

.92 

•94 

.96 

.98 

I.OO 

67 

24 

•85 

•87 

.88 

.90 

.91 

•93 

•94 

.96 

.98 

.00 

1.02 

1.04 

66 

25 

.89 

.90 

.92 

•93 

•95 

.96 

.98 

i  .00 

i  .02 

.04 

1.  06 

i.  08 

65 

26 

.92 

•93 

•95 

•97 

.98 

I.OO 

1.02 

1.04 

.06 

.08 

I.IO 

I.  12 

64 

27 
28 

•95 
08 

•97 

.98 

I.OO 

1.02 

1.04 

i.  05 

1.07 

.09 

.12 

1.14 

1.16 

ll 

•9° 

I.OO 

I  .O? 

•°3 

I.O5 

T  no 

1.07 

1  .09 

*' 

.20 

61 

29 
30 

1.05 

i  .03 
1.07 

i.  08 

•07 

.10 

i  .oy 
1.  12 

1.14 

1.16 

1:3 

.21 

.19 
•23 

1-25 

1.24 
1.28 

60 

31 

1.  08 

I.IO 

i.  ii 

•13 

1.  15 

.I? 

.20 

1.22 

.24 

.27 

1.29 

1.32 

59 

32 

i.  ii 

1.  13 

1-IS 

•17 

I.ig 

.21 

•23 

1.25 

.38 

•3° 

1.36 

58 

33 
34 

1.17 

1.  19 

I.  21 

•23 

1-25 

•27 

•30 

1.29 
1.32 

•31 
•35 

•34 
•37 

I-37 
1.40 

*'39 

56 

35 

1.20 

1.22 

1.24 

.26 

1.29 

•31 

•33 

1.36 

•38 

.41 

1.44 

1.47 

55 

36 

1.23 

1-25 

1.27 

•30 

1.32 

•34 

•37 

i-39 

.42 

•45 

1.47 

i-Si 

54 

37 

1.26 

1.28 

1.30 

•33 

i-35 

•37 

.40 

1.42 

•45 

.48 

1  54 

53 

38 

1.29 

1.31 

i-33 

•36 

1.38 

.40 

•43 

1.46 

.48 

•5* 

1.54 

1.58 

S2 

39 

1.32 

1.34 

1.36 

•39 

1.41 

•43 

.46 

1.49 

•52 

1.58 

i.6x 

40 

i-35 

i-37 

*-39 

.42 

1.44 

•47 

•49 

1.52 

•55 

•58 

1.61 

1.65 

50 

4i 

1-37 

1.40 

1.42 

•45 

1.47 

•50 

•53 

1-55 

•58 

.61 

1.64 

1.68 

49 

42 

1.40 

1.42 

i«45 

•47 

1.50 

•53 

•55 

.61 

.64 

1.68 

1.71 

48 

43 

1.48 

•50 

•56 

•58 

1.61 

.64 

.68 

1.71 

I-75 

47 

44 

i  '.46 

xij 

1.50 

•53 

i's6 

•58 

.61 

1.64 

.67 

,7i 

1.74 

1.78 

46 

45 

1.48 

1.51 

•S6 

1.58 

.61 

.64 

1.67 

.70 

•74 

1.77 

1.81 

45 

§299- 


TABLES. 


303 


FACTORS   FOR   THE    REDUCTION    OF   TRANSIT   TIME 
OBSERVATIONS. 

This  top  argument  is  the  star's  declination  ±  8. 


61*° 

62° 

62*° 

63= 

63i° 

64° 

<* 

65° 

65i° 

66° 

66*° 

67° 

4*6° 

1.56 

1.58 

1.61 

1.64 

1.67 

1.70 

1.74 

i.  80 

1.84 

44° 

47 

1.53 

1.56 

1.58 

1.61 

1.64 

1.67 

I.70 

1.76 

i.  80 

1.83 

1.87 

43 

48 

I-55 

1-58 

x.  60 

1.63 

1.66 

1.69 

1.72 

1.75 

1.79 

1.82 

1.86 

1.90 

42 

49 

1.58 

1.61 

1.63 

1.66 

1.69 

1.72 

1-75 

1.79 

1.82 

1.86 

1.89 

41 

5° 

i.  60 

1.63 

1.66 

1.69 

1.72 

1.78 

1.81 

1.85 

1.88 

1.92 

l:996 

40 

51 

1.63 

1.66 

1.68 

1.71 

i-74 

1.77 

i.  80 

1.84 

1.87 

1.91 

I-9S 

1.99 

39 

52 

1.65 

1.68 

1.71 

1.77 

I.  80 

1.83 

1.86 

1.90 

1.94 

1.98 

2.02 

38 

C| 

J 

53 

1.67 

1.70 

1.76 

1.79 

1.82 

1.85 

1.89 

1.93 

1.96 

2.00 

2.04 

37 

«3 

54 

1.69 

1.72 

I-75 

1.78 

1.81 

1.85 

1.88 

1.91 

i-95 

1.99 

2.03 

2.07 

36 

JJ 

I 

55 

1.72 

1.74 

1.77 

i.  80 

1.84 

1.87 

1.90 

1.94 

1.98 

2.OI 

2.05 

2.10 

35 

1 

c 

56 

1.74 

1.77 

i.  80 

1  13 

1.86 

1.89 

i.93 

1.96 

2.OO 

2.04 

2.08 

2.12 

34 

JO* 

•7. 

57 

1.76 

1.79 

1.82 

1.88 

I.9I 

i-95 

1.98 

2.  O2 

2.06 

2.10 

2.15 

33 

S- 

H 

58 

1.78 

1.81 

1.84 

I'.S* 

1.90 

1.93 

1.97 

2.01 

2.05 

2.08 

2.13 

2.17 

32 

J 

59 

i.  80 

1.83 

1.86 

1.89 

1.92 

1-95 

1.99 

2.03 

3.07 

2.  II 

2.15 

2.19 

EL 

^! 

60 

x.8i 

1.84 

1.88 

1.91 

1.94 

1.97 

2.01 

2.05 

2.09 

2.13 

2.17 

2.22 

3° 

n 

S 

61 

1.83 

1.86 

1.89 

i-93 

1.96 

2.00 

2.03 

2.07 

2.  II 

2.15 

2.19 

2.24 

29 

A 

rt 

62 

1.85 

1.88 

1.91 

1.94 

1.98 

2.01 

2.05 

2.09 

2.13 

2.17 

2.21 

2.26 

28 

g 

"^ 

63 

1.87 

1.90 

1.93 

1.96 

2.00 

2.03 

2.07 

2.  II 

2.15 

2.19 

2.23 

2.28 

27 

n 

"5 

64 

1.88 

x.gx 

i-95 

1.98 

2.  02 

2.05 

2.09 

2.13 

2.17 

2.21 

2.2S 

2.30 

26 

3 

S 

65 

1.90 

1.93 

1.96 

2.00 

2.03 

2.07 

2.  II 

2.14 

2.19 

2.23 

2.27 

2.32 

25 

"o 

3 

66 

67 

1.91 

1.95 
1.96 

1.98 
1.99 

2.01 
2.03 

"3 

2.08 
2.10 

2.12 
2.14 

2.16 
2.18 

2.20 
2.22 

2.25 
2.26 

2.29 
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2.34 
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1.97 

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2.04 

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2.19 

2.24 

2.28 

2.32 

2-37 

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1.96 

1.99 

2.  02 

2.06 

2.09 

2.13 

2.21 

2.25 

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2.03 

2  .40 

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2.05 

2.08 

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2.20 

2.24 

2.28 

2.32 

2-37 

2.42 

J9 

ET 

9 

72 

1.99 

2.03 

2.06 

2.09 

2.13 

2.17 

2.21 

2.25 

2.29 

2-34 

2.38 

2.43 

il 

£ 

4) 

•e 

73 
74 

2.00 

2.OI 

2.04 
2.05 

2.07 
2.08 

2.  II 
2.12 

2.14 
2.15 

2.l8 
2.19 

2.22 
2.23 

2.26 
2.27 

2.31 
2.32 

I'll 

2.40 
2.41 

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16 

to 

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2.02 

2.06 

2.09 

2.13 

2.16 

2.24 

2.29 

2-33 

2-37 

2.42 

2.47 

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2.03 

2.07 

2.10 

2.14 

2.17 

2.21 

2.25 

2.30 

2-34 

2-39 

2-43 

2.48 

M 

o 

o 

JE 

77 

2.04 

2.07 

2.  II 

2.15 

2.18 

2.22 

2.26 

2.31 

2-35 

2.40 

2.44 

2.49 

13 

** 

78 

2.05 

2.08 

2.12 

2.15 

2.19 

2.23 

2.27 

2.31 

2.36 

2.40 

2-45 

2.50 

12 

X 

1 

79 

2.06 

2.09 

2.13 

2.16 

2.  2O 

2.24 

2.28 

2.32 

2-37 

2.41 

2.46 

2.51 

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S 

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80 

2.06 

2.IO 

2.13 

2.17 

2.21 

2.25 

2.29 

2-33 

2.38 

2.42 

2.47 

2.52 

IO 

Bi 

2.07 

2.10 

2.14 

2.18 

2.21 

2.2$ 

2.29 

2-34 

2.38 

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2.IS 

2.18 

2.22 

2.26 

2.30 

2.34 

2-39 

2.43 

2.48 

2.53 

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83 

2.08 

2.12 

2.15 

2.19 

2.22 

2.26 

2.31 

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2-44 

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7 

84 

2.08 

2.12 

2.15 

2.19 

2.23 

2.27 

2.31 

2-35 

2.40 

2.45 

2-49 

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6 

85 

2.09 

2.  12 

2.l6 

2.19 

2.23 

2.27 

2.3I 

2.36 

2.40 

2-45 

2.50 

2-55 

5 

86 

2.09 

2.13 

2.16 

2.20 

2.24 

2.28 

2.32 

2.36 

2.41 

2-45 

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87 

2.09 

2.13 

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2.24 

2.28 

2.32 

2.36 

2.41 

2.46 

2.50 

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3 

88 

2.09 

2.13 

2.16 

2.20 

2.24 

2.28 

2.32 

2.36 

2.41 

2.46 

2.51 

2.56 

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89 

2.  IO 

2.13 

2.17 

2.  2O 

2.24 

2.28 

2.32 

2-37 

2.41 

2.46 

2.51 

2.56 

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00 

2.  IO 

2.13 

2.17 

2.  2O 

2.24 

2.28 

2.32 

2-37 

2.41 

2.46 

2.51 

2.56 

O 

The  bottom  line  on  this  page  is  the  collimation  factor  C  (=  sec  S). 


304 


GEODETIC  ASTRONOMY. 


§299- 


FACTORS    FOR  THE  REDUCTION  OF  TRANSIT  TIME 
OBSERVATIONS. 

This  top  argument  is  the  starts  declination  ±  5. 


«• 

68° 

68*° 

69° 

w 

70° 

H- 

70*° 

70*° 

71° 

f* 

71*° 

1° 

•°5 

•°5 

•°5 

•°5 

•°5 

•°5 

80° 

2 

.09 

.09 

.10 

.10 

.10 

.10 

.  10 

.  10 

.11 

.  II 

.  II 

.  II 

88 

O_ 

3 
4 

.18 

.19 

.19 

.20 

.20 

.20 

.21 

.21 

.21 

.21 

.22 

.22 

07 

86 

5 

•23 

•23 

•24 

•24 

•25 

•25 

.26 

.26 

.26 

.27 

•27 

•27 

85 

6 

•27 

.28 

.28 

•29 

•3° 

•31 

•31 

•31 

•32 

•32 

•33 

•33 

84 

7 

•32 

•33 

•33 

•34 

•35 

•36 

.36 

•37 

•37 

•37 

•38 

•38 

83 

8 

•36 

•37 

•38 

•39 

.40 

.41 

.41 

•42 

.42 

•43 

•43 

•44 

82 

9 

.41 

.42 

•43 

•44 

•45 

.46 

.46 

•47 

•47 

.48 

•49 

•49 

81 

10 

•45 

.46 

•47 

•49 

•50 

•51 

•51 

•52 

•53 

•53 

•54 

•55 

80 

ii 

12 

•50 
•54 

:S 

•52 

•57 

•53 
•58 

•54 
•59 

.56 
.61 

t 

•57 
.62 

•58 

•59 
.64 

$ 

.60 
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79 
78 

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•59 

.60 

.61 

•63 

.64 

.66 

.67 

.67 

.68 

.69 

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•71 

77 

14 

•63 

•  65 

.66 

.68 

.69 

.71 

.72 

.72 

•73 

•74 

•75 

•76 

76 

15 

.68 

.69 

•71 

.72 

•74 

.76 

•77 

•78 

.78 

•79 

.80 

.81 

75 

16 

•72 

•74 

•75 

•77 

•79 

.81 

.82 

•83 

.84 

•85 

.86 

•87 

74 

'7 

.76 

.78 

.80 

.81 

•83 

•85 

.86 

.88 

.89 

.90 

.91 

.92 

73 

18 

.81 

•83 

.84 

.86 

.88 

.90 

.91 

•93 

•94 

•95 

.96 

•97 

72 

19 

•  85 

•87 

.89 

•91 

•93 

•95 

.96 

.98 

•99 

1.  00 

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1.03 

20 

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.91 

•93 

•95 

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1.  00 

I.  01 

1.02 

1.04 

1.05 

i.  06 

i.  08 

70 

21 

•94 

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I'f 

1.  00 

.02 

1.05 

.06 

1.07 

1.09 

1.  10 

i.  ii 

69 
68 

23 

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1.09 

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1.14 

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I.  I7 

1.19 

1.20 

I  .21 

1.23 

67 

fif. 

24 
25 

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1.18 

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1.24 

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1.27 

1.28 

1.30 

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65 

26 

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1.22 

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1.28 

•30 

I  31 

1-33 

i-35 

I.36 

1.38 

64 

27 

.19 

.21 

.24 

1.27 

•30 

i  '33 

•34 

1.36 

1.38 

i-39 

63 

28 

•23 

•25 

.28 

1.31 

•34 

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I.4I 

1.42 

1.44 

i!46 

i!48 

62 

29 

•27 

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J-35 

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1.42 

•43 

1-45 

1.47 

1.49 

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J-53 

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1.39 

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1.46 

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1.50 

1.52 

1-54 

1.56 

1.58 

60 

31 

•35 

•38 

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1.44 

•47 

i-Si 

•52 

1-54 

1.56 

1.58 

i.  60 

1.62 

59 

32 

33 

•39 
.42 

.42 
•45 

•45 
•49 

1.48 

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•55 

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£ 

1.61 
1.65 

1.63 
1.67 

1.65 
1.69 

1.67 
1.72 

58 
57 

34 

.46 

•49 

•53 

i.  '56 

.60 

1.63 

.65 

lies 

1.70 

1.72 

1.74 

1.76 

56 

35 

•50 

•53 

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i.  60 

.64 

1.68 

.70 

1.72 

1.74 

1.76 

1.78 

1.81 

55 

36 

•54 

•57 

.60 

1.64 

.68 

1.72 

•74 

1.76 

1.78 

i.  80 

1.83 

1.85 

54 

37 

•57 

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•  64 

1.68 

•72 

1.76 

•78 

1.  80 

1.83 

1.85 

1.87 

1.90 

53 

38 

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1.72 

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i.  80 

.82 

1.84 

1.87 

1.89 

1.91 

1.94 

52 

39 

•65 

.68 

.72 

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1.84 

.86 

1.88 

1.91 

1.96 

1.98 

40 

.68 

•72 

•75 

1.79 

.84 

1.88 

.90 

i-93 

1.95 

1.97 

2.OO 

2.03 

50 

41 

•7i 

•75 

•79 

1.83 

.87 

1.92 

•94 

1.96 

1.99 

2.OI 

2.04 

2.07 

49 

42 

•75 

•79 

•83 

1.87 

.91 

1.96 

.98 

2.00 

2.03 

2.05 

2.08 

2.  II 

48 

43 

.78 

.82 

.86 

1.90 

•95 

i  99 

.02 

2.04 

2.07 

2.09 

2.12 

2.15 

47 

44 

.82 

-85 

Q- 

.90 

1.94 

.98 

2.03 

.06 

2.08 

2.  II 

2.13 

2.l6 

2.19 

46 

45 

1.85 

.09 

•93 

i  .97 

2.07 

.09 

2.23 

45 

§299- 


TABLES. 


30$ 


FACTORS   FOR  THE   REDUCTION   OF  TRANSIT  TIME 
OBSERVATIONS. 

This  top  argument  is  the  star's  declination  ±  5.  » 


w 

68° 

681° 

69° 

69*° 

70° 

*• 

70*. 

*• 

7i° 

*r 

*• 

46C 

1.88 

1.92 

1.96 

2.01 

2.05 

2.IO 

2.  I" 

...5 

2.18 

2.21 

2.24 

2.27 

44 

47 

1.91 

1.95 

2.OO 

2.04 

2.09 

2.14 

2.16 

.19 

2.22 

2.25 

2.27 

2.30 

42 

48 

1.94 

1.98 

2.02 

2.07 

2.  12 

2.17 

2.19 

.22 

2.2S 

2.28 

2.31 

2.34 

42 

49 

1.97 

2.01 

2.06 

2.  II 

2.16 

2.21 

2-23 

.26 

2.29 

2.32 

2.35 

2.38 

50 

2.OO 

2.04 

2.09 

2.14 

2.19 

2.24 

2.27 

.29 

2.32 

2-35 

2.38 

2.41 

4° 

51 

2.03 

2.07 

2.12 

2.17 

2.22 

2.27 

2.30 

•33 

2.36 

2-39 

2.42 

2-45 

39 

5= 

2.06 

2.10 

2.15 

2.20 

2.2S 

2.30 

2-33 

•  36 

2-39 

2.42 

2.45 

2.48 

38 

(-1 

<cT 

53 

2.09 

2.13 

2.18 

2.2| 

2.28 

2-33 

2.36 

•39 

2.42 

2-45 

2.48 

2.52 

37 

'r 

V 

54 

2.  II 

2.  16 

2.21 

2.26 

2.31 

2-37 

2-39 

42 

2-45 

2.48 

2.52 

2-55 

36 

o 

55 

2.14 

2.19 

2.23 

2.29 

2-34 

2.40 

2.42 

2-45 

2.48 

2.52 

2-55 

2.58 

35 

ft* 

C 

56 

2.17 

2.21 

2.26 

2.3I 

2-37 

2.42 

2-45 

2.48 

2-51 

2-55 

2.58 

2.61 

34 

2. 

*S; 

57 

2.19 

2.»4 

2.29 

2-34 

2-39 

2.45 

2.48 

2-51 

2-54 

2.58 

2.61 

2.64 

33 

~- 

Ji_ 

58 

2.22 

2.26 

2.31 

2-37 

2.42 

2.48 

2-51 

2-54 

2-57 

2.61 

2.64 

2.67 

32 

£ 

^J 

io1 

2.24 
2.26 

2.29 
8.31 

2-34 
2.36 

2-39 
2.42 

2-45 
2.47 

2-51 
2-53 

2-54 
2.56 

2-57 
2-59 

2.60 
2.63 

2.63 

2.66 

2.67 
2.69 

2.70 

2-73 

30 

"n 

C 
0 

61 

2.29 

2-33 

2-39 

2.44 

2.50 

2.56 

2-59 

2.62 

2.65 

2.69 

2.72 

2.76 

29 

0 

ri 

62 

2-31 

2.36 

2.41 

2.46 

2.52 

2.58 

2.6l 

2.64 

2.68 

2.71 

2.75 

2.78 

28 

C 
; 

a 

63 

2-33 

2.38 

2-43 

2.49 

2-54 

2.60 

2.64 

2.67 

2.70 

2.74 

2.77 

2.81 

27 

t 

H 

64 

2-35 

2.40 

2-45 

2-51 

2-57 

2.63 

2.66 

2.69 

2-73 

2.76 

2.80 

2.83 

26 

3 

£ 

65 

2-37 

2.42 

2.47 

2-53 

2-59 

2.65 

2.68 

2.71 

2-75 

2.78 

2.82 

2.86 

25 

0* 

8 

66 

2-39 

2.44 

2.49 

2-55 

2.61 

2.67 

2.70 

2.74 

2-77 

2.81 

2.84 

2.88 

24 

5" 

£ 

67 

2.41 

2.46 

2-51 

2-57 

2.63 

2.69 

2.72 

2.76 

2-79 

2.83 

2.86 

2.90 

23 

^ 

68 

2.42 

2.47 

2-53 

2-59 

2.65 

2.71 

2.74 

2.78 

2.81 

2.85 

2.88 

2.92 

22 

5 

1 

69 
70 

2.44 
2.46 

2-49 
2-51 

I'M 

2.61 
2.62 

2.67 
2.68 

2-73 
2-75 

2.76 
2.78 

2.80 
2.81 

2.83 
2.85 

2.87 
2.89 

2.90 
2.92 

21 
23 

i 

5 

3 

| 

7» 

2.47 

2.52 

2.58 

2.64 

2.70 

2-77 

2.80 

2.83 

2.87 

2.90 

2.94 

2.98 

I9 

f 

en 

72 

2.49 

2-54 

2-59 

2.65 

2.72 

2.78 

2.81 

2.85 

2.88 

2.92 

2.96 

3.00 

18 

i 

•i 

•c 

73 

2.50 

2-55 

2.6t 

2.67 

2-73 

2.80 

2.83 

2.86 

2.90 

2.94 

2-97 

3.01 

i7 

^ 

•s 

74 

2.51 

2-57 

2.62 

2.68 

2.81 

2.84 

2.88 

2.92 

2.95 

2-99 

3-03 

16 

a 

•£ 

75 

2.52 

2.58 

2.64 

2.70 

2.76 

2.82 

2.86 

2.89 

2-93 

2.97 

3-00 

3-04 

15 

- 

er 

76 

2-54 

2-59 

2.65 

2.71 

2.77 

2.84 

2.87 

2.91 

2-95 

2.99 

3-02 

3.06 

M 

.C 

77 

2-55 

2.60 

2.66 

2.72 

2.78 

2.85 

2.S8 

2.92 

2-95 

2.99 

3-03 

3.07 

13 

jj 

78 

2.56 

2.61 

2.67 

2-73 

2  79 

2.86 

2.89 

2-93 

2-97 

3-oo 

3-°4 

3-08 

12 

V 

I 

e 

2-57 
2-57 

2.62 
2.63 

2.68 
2.69 

2-74 
2-75 

2.80 

2.81 

2.87 
2.88 

2.91 
2.91 

2.94 
2-95 

2.98 
2.99 

3-02 
3-02 

3-05 
3-06 

3.09 
3.10 

II 
IO 

81 

2.58 

2.64 

2.69 

2.76 

2.82 

2.89 

2.92 

2.96 

3.00 

3-°3 

3-°7 

3-" 

9 

82 

2-59 

2.64 

2.70 

2.76 

2.83 

2.90 

2.93 

2-97 

3.00 

3-°4 

3-o8 

3-12 

8 

83 
84 

::£ 

2.65 
2.66 

2.71 
2.71 

2.77 
2.78 

2.83 
2.84 

2.90 
2.91 

2-94 
2.94 

2.97 
2.98 

3.01 
3-02 

3-°5 
3.06 

3-°9 
3-09 

3-13 
3-13 

7 
6 

85 

2.60 

2.66 

2.72 

2.78 

2.84 

2.91 

2-95 

2.98 

3.02 

3.06 

3.10 

3-14 

5 

86 

2.61 

2.66 

2.72 

2.78 

2.85 

2.92 

2-95 

2-99 

3-03 

3-06 

3.10 

S-M 

4 

87 

2.61 

2.67 

2.72 

2.79 

2.85 

2.92 

2-95 

2.99 

3-03 

3-°7 

3-  lx 

3 

88 

2.61 

2.67 

2.73 

2.79 

2.85 

2.92 

2.96 

2.99 

3-°3 

3.07 

1.  1  1 

3<J5 

2 

89 

2.61 

2.67 

2-73 

2.79 

2.86 

2.92 

2.96 

3.00 

3-°3 

3-07 

t  ii 

3-15 

I 

90 

2.61 

2.67 

2-73 

2-79 

2.86 

2.92 

2.96 

3.00 

3-°3 

3-07 

3-" 

3-15 

0 

The  bottom  line  on  this  page  is  the  collimation  factor  C  (=  sec  S). 


306 


GEODETIC  ASTRONOMY. 


§299. 


FACTORS    FOR   THE    REDUCTION   OF  TRANSIT  TIME 
OBSERVATIONS. 

This  top  argument  is  the  star's  declination  ±  S. 


7i*° 

72° 

72}° 

7*t° 

7*1° 

73° 

73i° 

73i° 

73i° 

74° 

74i° 

£ 

f 

•°S 

.06 

.06 

.06 

.06 

.06 

.06 

.06 

.06 

.06 

.06 

89° 

88 

2 

3 

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•17 

•17 

•  17 
.23 

.18 
•23 

.18 

.  24 

.18 
.24 

!i8 
.24 

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•*3 
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87 

86 

5 

.28 

.28 

•29 

.29 

.29 

•3° 

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•31 

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•32 

85 

6 

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•34 

•34 

•35 

•35 

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•37 

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84 

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•42 

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83 

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82 

F9 

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81 
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•  67 

.68 

.69 

.70 

79 

12 

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•  67 

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78 

13 

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•  75 

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•77 

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•  79 

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]4 

•77 

•78 

•79 

.80 

.82 

•83 

.84 

•  85 

.87 

.88 

.89 

76 

15 

•83 

.84 

•85 

.86 

•  87 

.89 

.90 

.91 

•93 

•94 

•95 

75 

16 

.88 

.89 

.91 

•  92 

•  93 

•94 

.96 

•97 

•99 

i  .00 

i.  02 

74 

*7 

•93 

•95 

.96 

•  97 

•99 

1.  00 

1.  01 

1.03 

1.05 

i.  06 

i.  08 

73 

18 

•99 

1.00 

i  .01 

1.03 
i.  08 

1.04 

i.  06 

i  .07 

1.09 

I.  10 

1.  12 

1.14 

72 

*9 

20 

i  .04 
1.09 

1-05 
i  .  ii 

i  .07 

1  .  12 

1.14 

i.iS 

1.17 

I'I3 

1.19 

1.20 

1.22 

1.24 

i.  '26 

71 

7o 

21 

i.U 

1.  16 

I.I7 

1.19 

I.  21 

1.22 

1.24 

1.26 

1.28 

1-30 

1.32 

69 

22 

1.20 

I.  21 

1.23 

1.25 

1.26 

1.28 

1.30 

1.32 

i-34 

1.36 

1.38 

68 

23 

1.25 

1.26 

1.28 

1.3° 

1.32 

i-34 

1.36 

1.38 

1.40 

1.42 

1.44 

67 

24 

1.30 

1.32 

i-33 

1-35 

1-37 

1-39 

1.41 

1-43 

i-45 

I.48 

1.50 

66 

25 

1-35 

i-37 

1.39 

i.4i 

1.42 

1-45 

1.47 

1.49 

i-5i 

i-53 

1.56 

65 

26 

1.40 

1.42 

1.44 

1.46 

1.48 

1.50 

1.52 

i-54 

i-57 

i-59 

1.61 

64 

27 
28 

*'45 
1.50 

i-47 
I-52 

1.49 
i'S4 

1.5- 
1-56 

l:ll 

*.& 

1.58 
1.63 

i.  60 

1.65 

1.62 
1.68 

1.65 
1.70 

1.67 
i-73 

63 
62 

29 
30 

*-S5 
i.  60 

1:11 

1:5 

1.61 

1.66 

:! 

1.66 
1.71 

1.68 
i-73 

1.71 
1.76 

i-73 
1.79 

1.76 
1.81 

1.79 
1.84 

61 

60 

3* 

1.64 

i.67 

1.69 

1.71 

1.74 

1.76 

1.79 

1.81 

1.84 

1.87 

1.90 

59 

32 

1.69 

i.7i 

1.74 

1.76 

1.79 

1.81 

1.84 

1.87 

1.89 

1.92 

i-95 

58 

33 

1.74 

1.76 

1.79 

x.8i 

i.84 

1.86 

1.89 

1.92 

i-95 

1.98 

2.01 

57 

34 

1.79 

1.81 

1.83 

1.86 

1.89 

1.91 

1.94 

1.97 

2.00 

2.03 

2.06 

56 

35 

1-83 

1.86 

1.88 

1.91 

1-93 

1.96 

1.99 

2.02 

2.05 

2.08 

2.  II 

55 

36 

1.88 

1.90 

i-93 

1-95 

1.98 

2.OI 

2.04 

2.07 

2.  IO 

2.  13 

2.16 

54 

37 

1.92 

i-95 

1.97 

2.00 

2.03 

2.06 

2.09 

2.12 

2-15 

2.l8 

2.22 

53 

38 

1.97 

1.99 

2.  O2 

2.05 

2.08 

2.  II 

2.14 

2.17 

2.  2O 

2.23 

2.27 

S2 

39 

2.OI 

2.04 

2.06 

2.09 

2.12 

2-15 

2.18 

2.22 

2.25 

2.28 

2.32 

5i 

40 

2.05 

2.08 

2.  II 

2.14 

2.17 

2.20 

2.23 

2.26 

2.30 

2-33 

2-37 

5° 

41 

2.09 

2.  12 

2.15 

2.18 

2.21 

2.24 

2.28 

2.31 

2.34 

2.38 

2.42 

49 

42 

2.14 

2.T6 

2.19 

2.22 

2.26 

2.»9 

2.32 

2.36 

2-39 

2-43 

2.46 

48 

43 

1.18 

2.21 

2.24 

2.27 

2.30 

2-33 

2-37 

2.40 

2.44 

2-47 

2.51 

47 

44 

2.22 

2.=5 

2.28 

2.31 

2-34. 

2.38 

2.41 

2-45 

2.48 

2.52 

2.56 

46 

45 

2.26 

2.2Q 

2.32 

2-35 

2.38 

2.42 

2-45 

2.49 

2-53 

2.56 

2.60 

45 

§299- 


TABLES. 


307 


FACTORS    FOR   THE   REDUCTION   OF   TRANSIT   TIME 
OBSERVATIONS. 

This  top  argument  is  the  star's  declination  ±  5. 


7'*° 

72° 

72F 

72*° 

*• 

73° 

73i° 

73*° 

73i° 

74° 

74i° 

* 

<, 

46° 

2.30 

2-33 

2.36 

2-39 

2.42 

2.46 

2.49 

2-53 

2-57 

2.61 

2.65 

44° 

47 

2-33 

2-37 

2.40 

2-43 

2.47 

2.50 

2-54 

2-57 

2.61 

2.65 

2.69 

43 

48 

2.37 

2.40 

2.44 

2-47 

2-51 

2-54 

2  .  58 

2.62 

2.66 

2.70 

2.74 

42 

49 
50 

2.41 

2-45 

2-44 

2.48 

2.48 
2.51 

2.51 
2-55 

2-55 

2.62 

2:66 

2.66 
2.70 

2.70 
2.74 

2.74 
2.78 

2.78 
2.82 

40 

5i 

2.48 

2.51 

2-55 

2.58 

2.62 

2.66 

2.70 

2.74 

2.78 

2.82 

2.86 

39 

52 

2.52 

2-55 

2.58 

2.62 

2.66 

2.69 

2-73 

2.77 

2.82 

2.86 

2.90 

38 

53 

2-55 

2.58 

2.62 

2.66 

2.69 

2-73 

2-77 

2.81 

2.85 

2.90 

2.94 

37 

54 

2.62 

2.65 

2.69 

2.73 

2.77 

2.81 

2.85 

2.89 

2-94 

2.98 

36 

55 

2.62 

2.65 

2.69 

2.72 

2.76 

2.80 

2.84 

2.88 

2.93 

2.97 

3  02 

35 

56 

2.65 

2.68 

2.72 

2.76 

2.80 

2.84 

2.88 

2.92 

2.96 

3.01 

3-05 

34 

57 

2.68 

2.71 

2-75 

2.79 

2.83 

2.87 

2.91 

2-95 

3-00 

3-04 

3.09 

33 

58 

.71 

2.74 

.78 

.82 

2.86 

2.90 

2.94 

2.99 

3-03 

3-08 

3.12 

59 
6o 

$ 

1:Z 

.81 
.84 

•85 
.88 

2.89 
2.92 

2-93 
2.96 

2.97 

3-02 

3-05 

3-o6 
3-09 

3-" 
3-M 

3-16 
3-19 

30 

61 

•79 

2.83 

.87 

.91 

2-95 

2-99 

3-°4 

3-08 

3-13 

3-17 

3-22 

29 

62 

.82 

2.86 

.90 

•94 

2.98 

3.02 

3-06 

3-" 

3.16 

3-20 

3-25 

28 

63 

.84 

2.88 

.92 

.96 

3-00 

3-°5 

3-°9 

3-18 

3-23 

3-28 

27 

64 

.87 

2.91 

•95 

•99 

3-03 

3-°7 

3.12 

3.16 

3-2C 

3-26 

3-31 

26 

65 

.89 

2-93 

•97 

3-01 

3-06 

3-io 

3-19 

3-24 

3-29 

3'34 

25 

66 

.92 

2.96 

3-oo 

3-04 

3-08 

3-13 

3-x7 

3-22 

3.27 

3-31 

3-37 

24 

67 

•94 

2.98 

3-02 

3-06 

3.10 

3-i5 

3-20 

3-24 

3-29 

3-34 

3-39 

23 

68 

.96 

3.00 

3-04 

3.08 

3-'3 

3-22 

3-26 

3-31 

3.36 

3-42 

22 

69 

.98 

3.02 

3.06 

3-10 

3-iS 

3.  19 

5-24 

3-29 

3-34 

3,39 

3-44 

21 

70 

3.00 

3-04 

3.08 

3.12 

3-'7 

3-21 

3-25 

3«3X 

3-4* 

3-46 

20 

7t 

3-02 

3.06 

3-^0 

3-14 

3-^9 

3-24 

3.28 

3-33 

3-38 

3-43 

3.48 

Ti 

72 

3-04 

3.08 

3.12 

3-16 

3-21 

3-25 

3-30 

3-35 

3-40 

3-45 

3-50 

18 

73 

3.05 

3-09 

3.14 

3.  18 

3-22 

3-27 

3-32 

3-37 

3-42 

3-47 

3-52 

74 
75 

3-07 

3.08 

3-" 

3-"5 

3.20 

3.21 

3-24 
3-26 

3-29 
3-30 

3-33 
3-35 

3-38 
3-4° 

3-44 
3-45 

3-49 
3-50 

3-54 
3-56 

16 
15 

76 

3.10 

3-M 

3-18 

3-23 

3-28 

3-32 

3-37 

3-42 

3'4l 

3-53 

3-58 

14 

77 

3*11 

3-iS 

3.19 

3-24 

3-29 

3-33 

3-38 

3-43 

3-48 

3-54 

3-59 

13 

79 

3.12 
3-i3 

1:5 

3.21 

3-22 

3.25 
3-26 

3-30 
3-31 

3-34 

3-39 

3:446 

3-49 
3-51 

3-55 

3.60 
3.62 

12 
II 

80 

3-19 

3-23 

3-27 

3-32 

3-37 

3-42 

3-47 

3-52 

3-57 

IO 

81 

3.15 

3-20 

3-24 

3.28 

3-33 

3.38 

3-43 

3-48 

3-53 

3-58 

3-64 

9 

82 

3.16 

3.20 

3-25 

3-29 

3-34 

3-39 

3-44 

3-49 

3-54 

3-59 

3.65 

8 

83 

3-21 

3.26 

3-3° 

3-35 

3-4° 

3-45 

3-49 

3-55 

3-66 

7 

3-j| 

3-22 

3-26 

3.31 

3-35 

3-40 

3-45 

3-50 

3-55 

3.61 

3-66 

6 

85 

3-22 

3-27 

3^31 

3.46 

3-51 

3-56 

3.61 

3-67 

5 

86 

3-19 

3-23 

3-27 

3-32 

3-36 

3-4i 

3.46 

3-Si 

3-57 

3-62 

3-68 

4 

87 

3-T9 

3-23 

3-28 

3-32 

3-37 

3-42 

3-47 

3-52 

3-57 

3-62 

3.68 

3 

88 

3-19 

3-23 

3-=8 

3-32 

3-37 

3-42 

3-47 

3-52 

3-57 

3.62 

3.68 

2 

89 

3-^9 

3-24 

3-28 

3-33 

3-37 

3-42 

3-47 

3-52 

3-57 

3-63 

3-68 

I 

90 

3-19 

3-24 

3-28 

3-33 

3-37 

3-42 

3-47 

3-52 

3-57 

3-63 

3-68 

O 

The  bottom  line  on  this  page  is  the  collimation  factor  C  (=  sec  5). 


308 


GEODETIC  ASTRONOMY. 


§299- 


FACTORS   FOR  THE  REDUCTION  OF  TRANSIT  TIME 
OBSERVATIONS. 

This  top  argument  is  the  star's  declination  ±  S. 


74*° 

74f° 

75° 

,* 

755° 

752° 

76° 

76*° 

76*° 

76*° 

77° 

77l° 

1 

1° 

.06 

.07 

.07 

.07 

•07 

.07 

.07 

.07 

.07 

.08 

.08 

.08 
.  16 

"& 

3 

.20 

.20 

.20 

.21 

.21 

.21 

.22 

.22 

.22 

•23 

•23 

•24 

87 

4 

.26 

.27 

.27 

•27 

.28 

.28 

.29 

.29 

•3° 

•30 

•32 

86 

5 

•33 

•33 

•34 

•34 

•35 

•35 

•36 

•37 

•37 

•38 

•39 

.40 

85 

6 

•39 

.40 

.40 

.41 

.42 

•42 

•43 

•44 

•45 

.46 

.46 

•47 

84 

e 

I 

9 

.46 

.46 
•53 

•47 
•54 
.60 

•48 
•55 
.61 

.50 

% 

I1 

•65 

•Si 

•67 

X 

.68 

•54 
.62 

•7° 

35 

•71 

83 
82 
81 

<[! 

10 

.65 

.66 

.67 

.68 

.69 

•  7i 

.72 

•73 

•74 

.76 

•77 

•79 

80 

C 

ii 

•71 

•73 

•74 

:2 

•?6 

'I7 

V7? 

.80 

.82 

•83 

<82 

.86 

79 

78 

y 

13 

•84 

.86 

.87 

.88 

.90 

.91 

•93 

•95 

•96 

.98 

1.  00 

1.02 

7° 
77 

sll 

M 

.91 

.92 

•94 

•95 

•97 

.98 

1.  00 

1.03 

1.04 

i.  06 

1.  08 

1.  10 

76 

•97 

.98 

I.  00 

i.  06 

1.02 

1.  08 

1.03 

1.05 

1.07 

1.09 

i.  TI 

•-«3 

i.i5 

1.  17 

75 

16 

1.03 

1.05 

1.  10 

I.  12 

1-25 

74 

rt 

3 

i.  09 
1.16 

1.17 

1.19 

1.  21 

1.23 

I-25 

1.28 

1-3° 

1.32 

i  .30 
i«37 

i  .32 
1.40 

73 
72 

•5 

19 

1.22 

1.24 

1.26 

1.28 

1.30 

1.32 

i-35 

1-37 

1.42 

1.47 

1 

20 

1.28 

1.30 

1.32 

i-34 

i-37 

i-39 

1.41 

1.44 

1.46 

1.49 

1.52 

i-55 

70 

rt 

21 

i-34 

1.36 

1.38 

1.41 

I-43 

1.46 

i  48 

1  51 

I-54 

1.56 

i-59 

1.62 

69 

u 

22 

1.40 

1.42 

1.50 

J-52 

1.58 

i  60 

1.63 

1.66 

1.70 

68 

£ 

23 

1.46 

1.49 

1.51 

i-54 

1.56 

1^62 

1.64 

1.67 

1.70 

i-74 

1.77 

67 

'c 

24 

i-55 

1-57 

i.  60 

1.63 

1.65 

1.68 

1.71 

1.74 

1.77 

x.8i 

1.84 

66 

6 

25 

I'.ll 

1.61 

1.63 

1.66 

1.69 

1.72 

1-75 

1.78 

1.81 

1.84 

1.88 

1.91 

65 

& 

26 

1.64 

1.67 

1.69 

1.72 

J'75 

1.78 

1.81 

1.84 

1.88 

1.91 

1.95 

1.99 

64 

rt 

27 

1.70 

1.78 

t.8i 

1.85 

1.88 

1.91 

1-95 

1.98 

2.02 

2.06 

63 

4J 

28 

1.76 

i!78 

Jill 

1.84 

1.87 

1.91 

1.94 

1.97 

2.OI 

2.05 

2.09 

2.13 

62 

"I 

29 

1.81 

1.84 

1.87 

1.90 

1.94 

1.97 

3.00 

2.04 

2.08 

2.  II 

2.15 

2.  2O 

61 

i 

3° 

1.87 

1.90 

»-93 

1.96 

2.00 

2.03 

2.07 

2.10 

2.14 

2.l8 

2.22 

2.27 

60 

4J 

31 

*-93 

1.96 

1.99 

2.02 

2.06 

2.09 

2.13 

2.17 

2.21 

2.25 

2.29 

2-33 

59 

32 

1.98 

2.01 

2.05 

2.08 

2.12 

2.15 

2.19 

2.23 

2.27 

2.31 

2.36 

2.40 

58 

33 

2.04 

3.07 

2.10 

2.14 

2.l8 

2.2T 

2.25 

2.29 

2-33 

2.38 

2.42 

2-47 

57 

5 

34 

2.09 

2.13 

2.16 

2.  2O 

2.23 

2.27 

2-31 

2-35 

2.40 

2-44 

2-49 

56 

JD 

35 

2.15 

2.18 

2.22 

2.25 

2.29 

2-33 

2-37 

2.4I 

2.46 

2.50 

2-55 

2.60 

55 

36 

2.  2O 

2.24 

2.27 

2.3I 

2-35 

2-39 

2-43 

2.47 

2.52 

2.56 

2.6l 

2.66 

54 

37 

2.25 

2.29 

2-33 

2.36 

2.40 

2-44 

2.49 

2-53 

2.58 

2.63 

2.67 

2-73 

53 

38 

2.30 

2-34 

2.38 

2.42 

2.46 

2.50 

2-55 

2-59 

2.64 

2.69 

2.74 

2-79 

52 

39 
40 

2-35 
2.40 

2-39 
2.44 

2-43 

2.48 

2-47 
2.52 

2.51 

2-57 

2.56 
2.6l 

2.60 

2.66 

2.65 
2.70 

2.70 
2-75 

2-75 
2.80 

2.80 
2.86 

2.85 
2.91 

50 

41 

2-45 

2.49 

2.53 

2.58 

2.62 

2.66 

2.71 

2.76 

2.8l 

2.86 

2.92 

2.97 

49 

42 

2.50 

2-54 

2.58 

2.63 

2.67 

2.72 

2.77 

2.81 

2.87 

2.92 

2-97 

3-°3 

48 

43 
44 

2:io 

2.63 

2.68 

2.68 
2-73 

2.73 

2-77 

2.77 
2.82 

2.82 
2.87 

2.87 

2.92 

2.92 
2.98 

2.98 
3-°3 

3-03 

3-09 
3.15 

3 

45 

2.65 

2  '.69 

2.73 

2.78 

2.82 

2.87 

2.92 

2.97 

3-°3 

3-i4 

3-20 

45 

§299. 


TABLES. 


309 


FACTORS    FOR   THE   REDUCTION    OF   TRANSIT   TIME 
OBSERVATIONS. 

This  top  argument  is  the  star's  declination  ±  5. 


74*° 

74f 

75° 

75i° 

75i° 

75*° 

*° 

* 

*• 

76|° 

77° 

77i° 

46» 

2.69 

2-73 

2.78 

2.82 

2.87 

2.92 

2-97 

3-03 

3-o8 

3-T4 

3-20 

3-26 

44° 

47 

2-74 

2.78 

2.83 

2.87 

2.92 

2.97 

3-02 

3.08 

3-13 

3.19 

3.25 

3-31 

43 

48 

2.78 

2.82 

2.87 

2.92 

2-97 

3-02 

3.07 

3-13 

3.18 

3-24 

3.3° 

3-37 

42 

49 
50 

2.82 
2.87 

2.87 
2.91 

2.92 
2.96 

2.96 
3.01 

3.01 
3.06 

3-07 
3." 

3-12 
3-17 

3-J8 

3-22 

1:3 

3-29 
3-34 

3-35 
3-41 

3-42 
3-47 

41 

4° 

51 

2.91 

2.95 

3.00 

3-05 

3.10 

3.16 

3.2I 

3.27 

3-33 

3-39 

3-45 

3-52 

39 

52 
53 

2.95 
2-99 

3.00 

3.04 

3-04 
3-09 

3-09 

3-iS 
3-*9 

3-20 

3-24 

3.26 
3-30 

3-31 

3.36 

3-38 
3.42 

3-44 
3-48 

3-50 
3-55 

3-57 
3.62 

38 
37 

cl 

54 

3-°3 

308 

3-13 

3.18 

3-23 

3.29 

3-34 

3.40 

3-47 

3-53 

3-60 

3-67 

36 

™ 

55 

3.07 

3-" 

3.16 

3-22 

3-27 

3.33 

3-39 

3-45 

3.5i 

3-57 

3.64 

3-71 

35 

r 

56 

3.10 

3-15 

3-20 

3.26 

3.31 

3-37 

3-43 

3.49 

3-55 

3.62 

3-68 

3.76 

34 

1 

H 
59 

3-T4 
3.21 

3.22 
3.26 

3.24 
3.28 
3.31 

3-33 
3-37 

3-35 
3-39 
3-42 

3-4i 

3-45 

3-47 
3-Si 

3-54 

3-53 

!:£ 

3.59 
3.63 
3-67 

3-66 
3-70 
3-74 

3-73 
3-77 

3.80 
3-84 
3-88 

33 
32 

I 

60 

3  24 

3-29 

3.35 

3.40 

3.46 

3.52 

3-58 

3.64 

3-78 

3*85 

3-92 

3° 

n 

$ 

61 

3.27 

3-33 

3.38 

3-44 

3-49 

3-55 

3.62 

3.68^ 

3-75 

3-82 

3.89 

3.96 

29 

62 
63 

3  3<> 

3-36 
3-39 

3.41 

3-44 

3-47 
3-50 

1:3 

3-59 
3.62 

3-65 
3-68 

3-72 

3-75 

3-82 

3-85 
3-89 

3-92 

4.00 
4.04 

28 
27 

1 

64 

3.36 

3-42 

3-47 

3'53 

3-59 

3.65 

3-72 

3.78 

3-85 

3-92 

4.00 

4.07 

26 

pi 

65 

3-39 

3-45 

3.50 

3.56 

3-62 

3.68 

3-75 

3-8i 

3-88 

3-95 

4-03 

4.11 

25 

cT 

66 

3-42 

3-47 

3-53 

3-59 

3-65 

3.7i 

3.78 

3-84 

3.91 

3-99 

4.06 

4«*4 

24 

5 

67 

3-44 

3-50 

3-56 

3.62 

3-68 

3-74 

3-87 

3-94 

4.02 

4.09 

4-17 

23 

68 

3-47 

3-53 

3-58 

3-64 

3-70 

3-77 

l'.*l 

3-90 

3-97 

4.05 

4.12 

4.20 

22 

5' 

69 

3-49 

3-55 

3-61 

3-67 

3-73 

3-79 

3-86 

3-93 

4.00 

4.07 

4-15 

4-23 

21 

P 

70 

3-52 
3  et 

3-57 

3  60 

3-63 

o.ge 

3-69 

3-75 

3  ?8 

3.82 
,  84 

3.89 

3-95 

4-03 

4.10 

4.18 

4.25 

2O 

5* 

p 

72 

•54 
3-56 

j.UO 

3-62 

3  P 
3-67 

3-74 

j./o 
3-8o 

I:S 

3-93 

4.00 

4.07 

4.23 

11 

« 

73 

3.58 

3.69 

3.76 

3.82 

3.89 

3-95 

4.02 

4.10 

4.17 

4-25 

4-33 

17 

5 

74 

3.6o 

3*65 

3.78 

3.84 

3-91 

3-97 

4.04 

4.12 

4.19 

4.27 

4.36 

16 

Sj 

75 

3.6! 

3-67 

3-73 

3-79 

3-86 

3-92 

3-99 

4.06 

4.14 

4-21 

4.29 

4-38 

15 

•^f 

76 

3-64 

3.69 

3-75 

3-82 

3-88 

3.94 

4.01 

4.08 

4.16 

4-23 

4-3' 

4.40 

14 

r> 

77 

3-65 

3-70 

3.76 

3-83 

3-89 

3.96 

4.03 

4.10 

4-17 

4-25 

4-33 

4-41 

13 

t/> 

78 

3-66 

3'72 

3.78 

3.84 

3.97 

4.04 

4.11 

4.19 

4-27 

4-35 

4-43 

12 

'us* 

79 

3.67 

3-73 

3-79 

3-86 

3-92 

3-99 

4.06 

4-13 

4.21 

4.28 

4.36 

4.45 

II 

0 

80 

3-68 

3-74 

3.87 

3-93 

4.00 

4.07 

4.14 

4.22 

4-3«> 

4.38 

4.46 

10 

^ 

Si 
82 

3-70 
3-71 

3-75 
3-76 

3.82 
3-83 

3-88 
3.89 

3-94 

4.01 

4.02 

4.08 
4-09 

4.16 
4-17 

4-23 
4-24 

4-32 

4-39 
4.40 

4.48 
4-49 

I 

83 

3-72 

3-77 

3-84 

3.90 

3-  '96 

4-°3 

4.10 

4.18 

4-25 

4-33 

4.41 

4-50 

7 

84 

3-72 

3.78 

3.84 

3.91 

3-97 

4.04 

4.11 

4.18 

4.26 

4-34 

4-42 

4-Si 

6 

85 

3-73 

3-79 

3-85 

3-98 

4-05 

4.12 

4.19 

4.27 

4-35 

4-43 

4-Si 

5 

86 

3-73 

3-79 

3.85 

3.92 

3.98 

4-05 

4.12 

4.20 

4.27 

4-35 

4-43 

4-52 

4 

87 

3-74 

3  79 

3-86 

3-92 

3-99 

4.06 

4.13 

4.20 

4.28 

4-36 

4-44 

4-52 

3 

88 

3-74 

3.80 
-,  g0 

3-86 

3  92 

3-99 

4.06 
4  06 

4.^3 

4.20 

4.28 

4.36 

4.44 

4.53 

2 

90 

3-74 
3-74 

3.00 
3.80 

3^86 

3-93 
3  93 

3*99 
3-99 

4-06 

4  •  *3 

4-13 

4.21 

4.28 

4.36 

4  »44 
4-44 

4  •  53 

4.53 

0 

The  bottom  line  on  this  page  is  the  collimation  factor  C  (=  sec  6). 


GEODETIC  ASTRONOMY. 


§299- 


FACTORS    FOR   THE    REDUCTION  OF  TRANSIT  TIME 
OBSERVATIONS. 

This  top  argument  is  the  star's  declination  ±  5. 


77i° 

77t° 

78° 

78}° 

*• 

78*° 

79° 

79*° 

79*° 

791° 

80° 

£ 

^ 

•• 

.08 

.08 

.08 

.09 

.09 

.09 

.09 

.09 

.10 

.10 

.10 

80° 

2 

.16 

.16 

.17 

•17 

.18 

.18 

.18 

.19 

.19 

.20 

.20 

88 

3 

.24 

•25 

•25 

.26 

.26 

•27 

•27 

.28 

29 

.29 

•3° 

87 

4 

•33 

•34 

•34 

•35 

•36 

•37 

•37 

•38 

•39 

.40 

86 

5 

.40 

.41 

.42 

•43 

•44 

•45 

.46 

•47 

.48 

•49 

•So 

85 

6 

•49 

•49 

•5i 

•5i 

•S2 

•54 

•55 

•S6 

•57 

•59 

.60 

84 

7 

.56 

•57 

•59 

.60 

.61 

.62 

.64 

•65 

.67 

.69 

.70 

83 

8 

.64 

.66 

.67 

.68 

.70 

•7i 

•73 

•75 

.76 

.78 

.80 

82 

9 

•72 

•74 

•75 

•77 

.78 

.80 

.82 

.84 

.86 

.88 

.90 

81 

10 

.80 

.82 

.84 

•85 

.87 

.89 

.91 

•93 

•95 

.98 

i  .00 

80 

ii 

.88 

.90 

•92 

•94 

•<* 

•98 

1.  00 

i  .02 

1.05 

.07 

.10 

79 

12 

.96 

•98 

1.  00 

i  08 

1.02 

1.09 

i.  ii 

1.14 

.20 

78 

M 

i  .04 

I  .  12 

I  .OO 

1.14 

I.  21 

1.27 

1.30 

i-33 

•36 

•3° 

•39 

76 

15 

1.20 

1.22 

1.25 

1.27 

1.30 

i-33 

1.36 

i-39 

1.42 

•  46 

•49 

75 

i6 

1.28 

1.30 

1-33 

1.35 

1.38 

1.41 

1.44 

1.48 

1.51 

•55 

•59 

74 

17 

j  .  75 

1.38 

1.40 

1.44 

1.47 

1.50 

1-53 

I-57 

i.  60 

.  64 

.68 

73 

18 

1-43 

1.46 

1.49 

1.52 

I-55 

1.58 

1.62 

1.66 

1.70 

•74 

.78 

72 

jg 

T'5X 

T-57 

I.  60 

1.63 

1.67 

1.71 

'•75 

1.79 

•83 

.87 

20 

1.58 

I'.ll 

1.65 

1.68 

1.72 

1.79 

1.83 

1.88 

.92 

•97 

70 

21 

1.65 

1.69 

1.72 

1.76 

i.  80 

1.84 

1.88 

1.92 

1.97 

.01 

.06 

69 

22 

1.77 

i.  80 

1.84 

1.88 

1.92 

1.96 

2.01 

2.06 

.  ii 

.16 

68 

23 

I'.li 

1.84 

1.88 

1.92 

1.96 

2.00 

2.05 

2.09 

2.14 

.20 

•25 

67 

24 

1.88 

1.92 

1.96 

2.OO 

2.04 

2.08 

2.13 

2.18 

2.23 

.29 

•34 

66 

25 

i-95 

1.99 

2.03 

2.07 

2.12 

2.17 

2.22 

2.27 

2.32 

•38 

•43 

65 

26 

2.02 

2.07 

2.  II 

2.15 

2.20 

2.25 

2.30 

2-35 

2.41 

.46 

•52 

64 

s 

2.10 
2.17 

2.14 

2.21 

2.l8 
2.26 

2.23 
2.31 

2.28 
2.36 

2-33 
2.41 

2.38 
2.46 

2-43 

2-49 

2.58 

•55 
.64 

.61 
.70 

63 
62 

29 

2.24 

2.28 

2-33 

2.38 

2-43 

2.48 

2-54 

2^60 

2.66 

•73 

•79 

61 

3° 

2.31 

2.36 

2.40 

2.46 

2.SI 

2.56 

2.62 

2.68 

2-74 

2.81 

2.88 

60 

31 

2.38 

2-43 

2.48 

2-53 

2.58 

2.64 

2.70 

2.76 

2.83 

2.89 

2-97 

59 

32 

2-45 

2.50 

2-55 

2.60 

2.66 

2.72 

2.78 

2.84 

2.91 

2.98 

3-05 

58 

33 

2.52 

2.57 

2.62 

2.67 

2-73 

2.79 

2.85 

2.92 

2-99 

3.06 

57 

34 

2.64 

2.69 

2-75 

2.80 

2.87 

2.93 

3-oo 

3-07 

3-14 

3.22 

56 

35 

I'.ls 

2.70 

2.76 

2.82 

2.88 

2.94 

3-o8 

3-iS 

3-23 

3-30 

55 

36 

2.72 

2.77 

2.83 

2.89 

2-95 

3.01 

3-08 

3-15 

3-23 

3-3° 

3.38 

54 

37 

2.78 

2.84 

2.90 

2.95 

3-02 

3-o8 

3-15 

3-23 

3-3° 

3.38 

3-47 

53 

38 

2.85 

2.90 

2.96 

3-02 

3-og 

3-16 

3-23 

3-3° 

3.38 

3-46 

52 

39 

2.91 

2-97 

3-°3 

3-09 

3.16 

3-23 

3-30 

3-37 

3-45 

3-53 

3-62 

51 

40 

2-97 

3-03 

3-09 

3-i6 

3.22 

3-29 

3-37 

3-45 

3-53 

3.61 

3-70 

5° 

4* 

3-°3 

3-°9 

3-16 

3.22 

3-29 

306 

3-44 

3-52 

3-6o 

3-69 

3-78 

49 

42 

3-°9 

3-22 

3-29 

3.36 

3-43 

3-51 

3-59 

3-67 

3.76 

3-85 

48 

43 

3.21 

3-28 

3-35 

3-42 

3-50 

3-66 

3-74 

3-83 

3-93 

47 

44 

3-21 

3-27 

3-34 

3.41 

3.48 

3-56 

3-64 

3-72 

3.81 

3-91 

4.00 

46 

45 

3-27 

3-33 

3-40 

3-47 

3-55 

3.62 

3-71 

3-79 

3-88 

3-97 

4.07 

45 

TABLES. 


FACTORS  FOR  THE  REDUCTION  OF  TRANSIT  TIME 
OBSERVATIONS. 

This  top  argument  is  the  star's  declination  ±  «. 


774° 

77l° 

78° 

78*° 

78*° 

78*° 

79° 

79i° 

79i° 

HT 

80° 

t 

, 

46° 

47 

3-32 
3.38 

3-39 
3-45 

3.46 
3-52 

3-53 
3-59 

3  '-67 

3.69 

3  75 

3-77 
383 

3-86 

3-95 

4.04 

4.14 

44° 
43 

48 
49 

3-43 
3-49 

3-50 

3-57 
3-63 

3-71 

3-73 
3-79 

3-8r 
3-87 

3-89 
3.96 

3-98 
4-05 

4.14 

4.24 

4-35 

50 

3-54 

3-6i 

3-68 

3.76 

3-84 

3-93 

4.02 

4.11 

4.20 

4-30 

4.41 

40 

Si 

3-e9 

3-66 
3  •  7* 

3-74 
3"  79 

3-82 

0.87 

3-90 

3ne 

3.98 

4.07 

4-17 

4.26 

4-37 

.48 

39 

_6 

53 

3*69 

3-77 

3.84 

o*  w/ 

3-92 

•yj 

4.01 

4.09 

4.19 

4.28 

4-38 

4-49 

iio" 

3° 
37 

54 

55 

3-74 
3-78 

3.81 
3-86 

3-89 

3-94 

3-97 

4.02 

4.o6 
4.11 

4-i5 
4.20 

4.24 
4.29 

4-34 
4-39 

4-44 
4-50 

4-55 
4.60 

.66 

.72 

36 
35 

56 

3-83 

3.91 

3-99 

4.07 

4.16 

4-25 

4-34 

4.44 

4-55 

4.66 

4-77 

34 

57 

3.88 

3-95 

4.04 

4.12 

4.21 

4-3° 

4-39 

4-50 

4.60 

4.72 

4-83 

33 

58 

3-92 

4.00 

4.08 

4.16 

4-25 

4-35 

4.44 

4-55 

4-65 

4-77 

4.88 

B 

3.96 
4.00 

4.04 
4.08 

4.12 
4.17 

4.21 
4-25 

4-30 

4-34 

4-39 
4.44 

•49 

•54 

eg 

4.60 
4.64 

4.70 
4-75 

A.  80 

4.82 
4-87 

4.94 
4-99 

3T 
30 

ori 

62 

4.08 

4.16 

4.25 

4-34 

4  "39 
4-43 

"53 

t  .  50 

.63 

4-73 

4-00 
4.85 

4.92 
4.96 

1:3 

29 
28 

63 

4.12 

4.20 

4.29 

4-38 

4-47 

•57 

.67 

4.78 

4.89 

5-oi 

55ll 

26 

65 

4.19 

4.27 

4.36 

4-45 

4-55 

•65 

•75 

4.86 

4-93 
4-97 

5-°5 
5-09 

5.10 

5-22 

25 

66 

4.22 

4-3i 

4.40 

4-49 

4.58 

.68 

•79 

4.90 

S.oi 

S-I4 

5.26 

24 

67 

4.26 

4-34 

4.43 

4-52 

4.62 

.72 

.82 

4-94 

5-°S 

5-18 

5-3° 

23 

68 

4.28 

4-37 

4.46 

4-55 

4-65 

•75 

.86 

4-97 

5-09 

5-21 

5-34 

22 

69 

4-32 

4.40 

4.49 

4-58 

4.68 

•79 

4.89 

S-oo 

5.12 

516 

5-25 
5.28 

5.38 

21 

70 
71 

4-34 

4-37 

4-43 

4.46 

4-55 

4.64 

4-74 

4-85 

4  '93 

5-<>7 

•  IU 

5-19 

5-32 

S-41 

5-45 

r 

72 

4-39 

4-48 

4-57 

4.67 

4-77 

4.88 

4.98 

5.22 

5-34 

5.48 

18 

73 

4.42 

4-51 

4.60 

4.70 

4.80 

4.90 

5.01 

5-13 

5-25 

5-37 

*7 

74 

4-44 

4-53 

4.62 

4.72 

4.82 

4-93 

5-°4 

5-  T5 

5-27 

5-4° 

5-53 

16 

75 

4.46 

4-55 

4-65 

4-74 

4.84 

4-95 

5-06 

5-30 

5-43 

5.56 

15 

76 

4.48 

4-57 

4.67 

4.76 

4.87 

4  -97 

S-°9 

5-20 

5-32 

5-45 

5-59 

14 

77 

4-50 

4-59 

.68 

4.78 

4.89 

4-99 

5-" 

5-22 

5-35 

5-47 

5.61 

13 

78 

4-52 

4.61 

.70 

4.80 

4.91 

5.01 

5-13 

5-24 

5-37 

5-50 

5-63 

12 

79 

4-54 

4-63 

.72 

4.82 

4.92 

5-03 

5-14 

5.26 

5-39 

5-52 

5.65 

II 

80 

4-55 

4.64 

•74 

4.84 

4-94 

5-°S 

5-i6 

5-28 

5-4° 

5-54 

5-67 

10 

Si 

4.56 

4-65 

•75 

4-85 

4-95 

5-o6 

5.18 

5-3° 

5-42 

5-55 

5-69 

9 

82 

4-57 

4.67 

i  .76 

4.86 

4-97 

5-o8 

5-*9 

5.-  3* 

5-43 

5.56 

5-70 

8 

83 

4-59 

4.68 

.78 

4.87 

4-98 

5-09 

5-20 

5-32 

5-45 

5.58 

5-72 

7 

84 

4.60 

4-69 

•79 

4.88 

4.99 

5-21 

5'33 

5-46 

5-59 

5-73 

6 

85 

4.60 

4.69 

•79 

4.89 

5-00 

5-" 

5-22 

5-34 

5-47 

S-6o 

5-74 

5 

86 

4.61 

4.70 

.80 

4.90 

5-oo 

5-" 

5-23 

5-35 

5-47 

5-6i 

5-74 

4 

87 

4.62 

4-71 

.81 

4.90 

5.01 

5-12 

5-23 

5-35 

5-48 

5.61 

5-75 

3 

88 
go 

4.62 

4  62 

4-71 

.81 

4-91 

5-oi 

SOI 
•  *•"• 

5-12 

5.12 

5-24 

5  ***4 

5-36 

5.48 
5  *49 

5.6. 

5.62 

5-75 

2 

I 

°y 
90 

4-"-* 
4.62 

4.71 

4.81 

4.91 

5-02 

5.13 

5-24 

5^6 

5-49 

5-62 

SO'6 

0 

The  bottom  line  on  this  page  is  the  collimation  factor  C  (=  sec  S). 


3I2 


GEODETIC  ASTRONOMY. 


§300. 


300.     FACTORS   FOR   THE   REDUCTION    OF  TRANSIT  TIME 

OBSERVATIONS   AT   CORNELL   UNIVERSITY. 

<t>  =  42°  27'. 


a 

v4 

B 

A11  + 

C 

All  4-  for 
lamp  west, 
and  —  for 
lamp  east. 

' 

4 

B 

AH  4 

C 

All  4-  for 
amp  west, 
and  —  for 
lamp  east. 

—  40° 

—  35 

4-  1.29 
+  1.19 

0.17 
0.26 

1.31 

1.22 

4-62*° 
63 

—  -74 

—  -77 

:3 

.20 

—  3° 

4-  1  .10 

o-35 

15 

63* 

—  .Bo 

.09 

•24 

—  25 

20 

tl.02 
0.94 

0.42 
0.49 

IO 

06 

64 

64* 

~  '%7 

.12 

.28 
•32 

IO 

4-0.87 

4-0.80 

0.56 
0.62 

04 
02 

65 

65* 

—  -91 
—  -94 

.19 
.22 

•37 
•  41 

—  8 

4-0.78 

0.64 

01 

66 

—  -98 

•25 

.46 

—  6 

4-0.75 

0.67 

OI 

66* 

—  .02 

•29 

•51 

—  4 

4-0.73 

0.69 

00 

67 

—  .06 

•33 

•56 

—  2 

4-0.70 

0.71 

oo 

67* 

—   .10 

•37 

.61 

0 

4-  2 

--0.67 
--  0.65 

0.74 
0.76 

00 

oo 

68 

68* 

—  .20 

.41 
•  45 

.67 
•73 

4 

--  0.62 

o.79 

00 

69 

—  -25 

.50 

•79 

6 

-  -  o  .  60 

0.81 

01 

69* 

—  -30 

•54 

.86 

8 

--  o-57 

0.83 

01 

7° 

—  '35 

•59 

•92 

0 

4-0.54 

0.86 

02 

-  -38 

.62 

.96 

12 

0.88 

.02 

70* 

—  -41 

.64 

3.00 

14 

4-0.49 

0.91 

•03 

7°* 

—  -44 

.67 

3-03 

16 

20 

4-o.46 
4-0.43 
4-0.41 

o-93 
0.96 
0.98 

•04 

a 

5 

—  -47 
—  -SO 
—  -53 

.70 

3-07 
3-" 
3-15 

22 

-4-0.38 

I.  01 

.08 

71* 

-  .56 

•79 

3-19 

3 

+  0.35 
4-  0.32 

1.04 
1.07 

.09 

.11 

72 
72* 

—  .60 
-  .63 

.82 
•85 

3-24 
3-28 

28 

4-0.28 

1.  10 

•  13 

72* 

—  .66 

.88 

3-33 

3° 

4-  0-25 

1-13 

•15 

72* 

—  .70 

.91 

3-37 

S2 

4-  O.22 

!.I6 

.18 

73 

—  -74 

•95 

3-42 

-j-o.i8 
4-o-i4 

1.19 
1.23 

.21 
.24 

73* 
73* 

—  .78 
—  .82 

3-02 

3-47 
3-52 

38 

4-  o.io 

1.27 

.27 

73* 

—  .86 

3  '05 

3-57 

40 

-j-o.  06 

1.30 

.31 

74 

—  .90 

3-09 

3-63 

42 

4-0-03 
4-  o.oi 

1.32 

i  -35 

•33 
•35 

74* 
74* 

3-13 
3-17 

3-68 
3-74 

43 

—  O.OI 

i  37 

•37 

74* 

—  03 

3-21 

3-8o 

44 

—  0.04 

i-39 

•39 

75 

—  .08 

3-26 

3-86 

45 

—  0.06 

.41 

75* 

—   13 

3.30 

3-93 

46 
47 

—  0.09 

—  O.  12 

!'.  Jo 

•44 
•47 

1 

-  .18 
—  -23 

3-35 
3-40 

3-99 
4.06 

48 

—  0.14 

1.49 

•49 

76 

-  .28 

3-45 

4-13 

49 

—  0.17 

i-Si 

•S2 

76* 

—  -34 

3-50 

4.21 

5° 

—  0.20 

•  •54 

•56 

76* 

—  .40 

3-55 

4.28 

51 

—  0.24 

76* 

_  .46 

3-6o 

4-36 

S2 

—  0.27 

i  .60 

.62 

77 

—  -52 

3-66 

4-44 

53 
54 

—  0.30 
—  0.34 

1.63 
1.67 

.66 
.70 

i 

9 

-0.38 
—  0.42 

1.70 
1.74 

•74 
•79 

77* 
78 

—  .72 
—  .80 

3-85 
3-91 

JiSJ 

ii 

-o.Je 
—  0.51 

It 

£ 

78* 

-  -87 
—  -95 

3-98 
4.06 

4  9* 

5-02 

59 

—  0.55 

1.86 

•  94 

78* 

—  3-03 

4-13 

5-13 

60 

—  0.60 

1.91 

.00 

79 

4.21 

5-24 

60* 
61 

—  0.63 

—  0.66 

1.93 
1.96 

a 

i 

—  3-21 

4.29 
4-38 

5.36 
5-49 

61  J 

—  0.68 

1.98 

.  IO 

79* 

—  3-41 

4-47 

5-62 

+  62 

—  0.71 

2.01 

•  13 

4-8o 

—  3-51 

4-57 

5.76 

This  table  is  computed  for  latitude  42°  27'.  It  may  be  used,  however,  for  any  station 
whose  latitude  does  not  differ  from  that  value  by  more  than  7'.  For  a  station  in  latitude 
42°  20',  or  in  latitude  42°  34',  the  maximum  error  in  the  table  is  two  units  in  the  last  place. 


§303- 


TABLES. 


313 


301.     CORRECTION    TO    TRANSIT  OBSERVATIONS    FOR 
DIURNAL   ABERRATION. 

The  correction  is  negative  when  applied  to  observed  times,  except  for  sub-polars.    (See  §  96.) 


Declination  =  5. 

Latitude 

=  *. 

o° 

10° 

20° 

30° 

40° 

50° 

60° 

70° 

80° 

o° 

0».02 

0*.02 

0*.02 

O8.O2 

o*.o3 

o8.  03 

o8.  04 

o*.o6 

0*.I2 

IO 

0  .02 

o  .02 

0  .02 

o  .02 

o  .03 

o  .03 

o  .04 

0.06 

0.12 

20 

o  .02 

o  .02 

0  .02 

0  .02 

o  .03 

o  .03 

o  .04 

o  .06 

0.12 

30 

o  .02 

o  .02 

o  .02 

o  .02 

o  .02 

o  .03 

o  .04 

o  .05 

O  .11 

40 

o  .02 

0  .02 

o  .02 

o  .02 

o  .02 

o  .03 

o  .03 

o  .05 

o  .09 

50 

O  .OI 

0  .01 

O  .OI 

o  .02 

o  .02 

o  .02 

o  .03 

0.04 

0.08 

60 

0  .01 

0  .01 

0  .01 

O  .OI 

0  .01 

0  .02 

o  .02 

o  .03 

o  .06 

70 

0  .01 

0  .01 

0  .01 

0  .01 

0  .01 

0  .01 

o  .or 

o  .02 

o  .04 

80 

o  .00 

o  .00 

0  .00 

0  .00 

o  .00 

o  .01 

o  .01 

o  .01 

o  .02 

302.     RELATIVE   WEIGHTS    FOR   TRANSIT    OBSERVATIONS 
DEPENDING   ON    THE   STAR'S    DECLINATION.     (See  §  m.) 

s 

iu 

Vw 

£ 

•w 

y-w 

a 

TV 

Vw 

o 

IO 
20 
30 
40 

I.OO 

0.99 

0.95 
0.87 
o.  76 

I.OO 

0.99 
0.97 

o-93 
0.87 

45 
50 

55 
60 

65 

0.69 
0.61 
0.52 
0.42 
0-33 

0.83 
0.78 
0.72 
0.65 

0.57 

70 

75 
80 

85 

O.24 
0.14 
0.07 

0.02 

0-49 

0.37 
0.26 
0.14 

.In  the  application  of  the  multiplier  \'~^  it  generally  suffices  to  employ 
but  one  significant  figure. 

303.     RELATIVE   WEIGHTS    FOR    INCOMPLETE    TRANSITS. 

(See  §  i  IT.) 

For  eye  and  ear  observations. 

For  observations  with  a  chronograph. 

No.  of 
Lines 

5  Lines  in 
Reticle. 

7  Lines  in 
Reticle. 

9  Lines  in 
Reticle. 

ii  Lines  in 
Reticle. 

13  Lines  in 
Reticle. 

Obs. 











w 

Vw 

w 

MIV 

•w 

y-iv 

IV 

4/w 

IV 

Vw 

I 

0.40 

0.63 

0.36 

0.60 

0.42 

0.65 

6.42 

0.65 

0.41 

0.64 

2 

0.64 

0.80 

0-57 

0-75 

0.62 

0.79 

0.62 

0.79 

O.6O 

0.77 

3 

0.80 

0.89 

0.71 

0.84 

o.73 

0.85 

0.73 

0.85 

0.71 

0.84 

4 

0.92 

0.96 

0.82 

O.gi 

0.82 

0.91 

0.81 

0.90 

0.79 

0.89 

5 

I.OO 

I.OO 

0.90 

0-95 

0.87 

0.93 

0.86 

o.93 

0.84 

0.92 

6 

0.95 

0.97 

0.92 

0.96 

0.90 

o.95 

0.88 

0.94 

7 

I.OO 

I.OO 

0-95 

0.97 

0-93 

0.96 

0.91 

0.95 

8 

0.98 

0-99 

0.95 

0.97 

0-93 

0.96 

9 

I.OO 

I.OO 

0.97 

0.98 

0-95 

0.97 

10 

0.99 

o.99 

0-97 

0.98 

ii 

I.OO 

I.OO 

0.98 

0.99 

12 

0.99 

0.99 

13 

I.OO 

I.OO 

GEODETIC  ASTRONOMY. 


304. 


304,     CORRECTION    TO    LATITUDE    FOR    DIFFERENTIAL 
REFRACTION. 

The  sign  of  the  correction  is  the  same  as  that  of  the  micrometer  difference 
(See  §  150.) 


i  Diff  .  of 
Zenith 
Distances. 

Zenith  Distance. 

0° 

10° 

20° 

25° 

3°° 

35° 

40° 

45° 

o'.o 

o".oo 

o".oo 

o".oo 

o".oo 

o".oo 

o".oo 

o".oo 

o".oo 

0.5 

0    .01 

0    .01 

o  .01 

o  .01 

0    .01 

o  .01 

o  .01 

0    .02 

I  .O 

o  .02 

0    .02 

0   .02 

o  .02 

0    .02 

0    .02 

o  .03 

o  .03 

I  -5 

o  .03 

o  .03 

o  .03 

o  .03 

o  .03 

o  .03 

o  .04 

o  .05 

2  .0 

o  .03 

o  .03 

o  .04 

o  .04 

o  .04 

o  .05 

o  .06 

o  .07 

2-5 

o  .04 

'  o  .04 

o  .05 

o  .05 

o  .05 

o  .06 

o  .07 

o  .08 

3-0 

o  .05 

o  .05 

o  .06 

o  .06 

o  .07 

o  .08 

o  .09 

o  .10 

3.5 

o  .06 

o  .06 

o  .07 

o  .07 

o  .08 

o  .09 

o  .10 

0    .12 

4.0 

o  .07 

o  .07 

o  .08 

o  .08 

o  .09 

o  .10 

O    .11 

o  .13 

4.5 

o  .08 

o  .08 

o  .09 

o  .09 

0    .10 

0    .11 

o  .13 

o  .15 

5-0 

o  .08 

o  .09 

0    .10 

o  .10 

0    .11 

o  .13 

o  .14 

o  .17 

5  -5 

o  .09 

0    .10 

0   .10 

O    .11 

0    .12 

o  .14 

o  .16 

o  .18 

6.0 

0    .10 

0    .10 

0    .11 

O    .12 

o  .13 

o  .15 

o  .17 

o  .20 

6.5 

0    .11 

0    .11 

0    .12 

o  .13 

o  .14 

o  .16 

o  .19 

O   .22 

7-0 

0    .12 

0    .12 

o  .13 

o  .14 

o  .15 

o  .18 

0    .20 

o  .24 

7-5 

o  .13 

o  .13 

o  .14 

o  .15 

o  .16 

o  .19 

O    .21 

o  .25 

8.0 

o  .13 

o  .14 

o  .15 

o  .16 

o  .18 

O    .21 

o  .23 

o  .27 

8.5 

o  .14 

o  .15 

o  .16 

o  .17 

o  .19 

O   .22 

o  .24 

o  .29 

9.0 

o  .15 

o  .16 

o  .17 

o  .18 

0    .20 

o  .23 

o  .26 

o  .30 

9-5 

o  .16 

o  .17 

o  .18 

0   .20 

0    .21 

o  .24 

o  .27 

o  .32 

10  .0 

o  .17 

o  .18 

o  .19 

0    .21 

o  .23 

o  .26 

o  .29 

o  -34 

10.5 

o  .18 

o  .19 

0    .20 

O    .22 

o  .24 

o  .27 

o  .30 

o  .35 

II  .0 

o  .18 

o  .19 

0    .21 

o  .23 

o  .25 

o  .28 

o  .31 

o  .37 

ii  .5- 

o  .19 

0    .20 

0    .22 

o  .24 

o  .26 

o  .30 

o  -33 

o  -39 

12  .0 

0    .20 

0    .21 

o  .23 

o  .25 

o  .27 

o  .31 

o  -34 

o  .40 

12.5 

O    .21 

0    .21 

o  .24 

o  .26 

o  .28 

o  .32 

o  .36 

o  .42 

13.0 

O   .22 

0    .22 

o  .25 

o  .27 

o  .29 

o  .33 

o  .37 

o  .44 

13.5 

o  .23 

o  .23 

o  .26 

o  .28 

o  .30 

o  -34 

o  .39 

o  .45 

14.0 

o  .23 

o  .24 

o  .27 

o  .29 

o  .31 

o  -35 

o  .40 

o  .47 

14.5 

o  .24 

o  .25 

o  .28 

o  .30 

o  .32 

o  .36 

o  .41 

o  .49 

15-0 

o  .25 

o  .26 

o  .28 

o  .31 

o  .34 

o  .38 

o  -43 

o  .50 

15.5 

o  .26 

o  .27 

o  .29 

o  .32 

o  -35 

o  -39 

o  .44 

o  .52 

16.0 

o  .27 

o  .28 

o  .30 

o  -33 

o  .36 

o  .40 

o  .46 

o  .54 

16.5 

o  .28 

o  .29 

o  .31 

o  .34 

o  -37 

o  .41 

o  .47 

o  .55 

17.0 

o  .28 

o  .29 

o  .32 

o  .35 

o  .38 

o  .42 

o  .49 

o  -57 

17.5 

o  .29 

o  .30 

o  -33 

o  .36 

o  -39 

o  .44 

o  .50 

o  .59 

18  .0 

o  .30 

o  .31 

o  -34 

o  .37 

o  .40 

o  -45 

o  .52 

o  .60 

18.5 

o  .31 

o  .32 

o  -35 

o  .38 

o  .41 

o  .46 

o  .53 

o  .62 

19  .0 

o  .32 

o  -33 

o  .36 

o  .39 

o  .43 

o  .48 

o  .54 

o  .64 

19.5 

o  .33 

o  -34 

o  .37 

o  .40 

o  .44 

o  .49 

o  .56 

o  .66 

20.0 

o  .34 

o  -35 

o  .38 

o  .41 

o  -45 

o  .50 

o  .57 

o  .67 

§306. 


TABLES. 


315 


305.     CORRECTION    TO    LATITUDE   FOR   REDUCTION    TO 
MERIDIAN. 

The  sign  of  the  correction  to  the  latitude  is  positive  except  for  stars  of  negative  declination 
(south  of  the  equator).    (See  §  151.) 


Hour-angle. 

s 

10" 

is9 

20s 

25' 

30' 

35* 

40' 

45* 

50" 

55* 

6o> 

£ 

0° 

5 

o".oo 

0    .00 

o".oo 

0    .01 

o".oo 

0    .01 

o".oo 

0    .01 

o".oo 

0    .02 

o".oo 
o  .03 

o"  .  oo  o"  .  oo 
o  .040  .05 

o".oo 
o  .06 

o".oo 
o  .07 

o".oo 
o   .09 

90° 

85 

10 

o  .oolo  .01 

o  .02 

o  .03 

o  .04 

o  .060  .08 

o  .09 

0    .12 

o  .14 

o   .17 

80 

15 

0    .01 

0    .02 

o  .03 

o  .04 

o  .06 

o  .08 

0    .11 

o  .14 

o  .17 

O    .21 

o  .24 

75 

20 

0    .01 

o   .02 

o  .04 

o  .05 

o  .08 

O    .  II 

o  .14 

o  .18 

0    .22 

o   .27 

o   .32 

70 

25 

o  .01 

o  .02 

o  .04 

o  .07 

o  .09 

o  .13 

o  .17 

0    .21 

0    .26 

o  .32 

o  .38 

65 

30 

o  .01 

o  .03 

o  .05 

o  .07 

0    .11 

o  .140  .19 

o  .24 

o  .30 

o  .36 

o  .42 

60 

35 

0    .01 

o  .03 

o  .05 

o  .08 

0    .12 

o  .16 

0    .21 

0    .26 

o  .32 

o   -39 

o  .46 

55 

40 
45 

0    .01 
0    .01 

o  .03 
o  .03 

o  .05 
o  .06 

o  .08 
o  .08 

6    .12 
0.12 

0    .160    .22 
0    .1710    .22 

o  .27 

0    .28 

o  .340   .41 
o  .340  .41 

o  .48 
o  .49 

50 
45 

306.      CORRECTION    FOR    CURVATURE    OF    APPARENT    PATH 
OF    STAR,   IN   MICROMETER  VALUE   DETERMINATIONS. 

The  correction  as  tabulated  is  £(15  sin  i")7Ts  —  ^(15  sin  i")4Ta. 


Apply  the  corrections  given  in  the  table  directly  to  the  observed  chronometer  times,  adding 
them  before  either  elongation  to  the  times,  and  subtracting  them  after  either  elongation. 

(See  §  159.) 


T 

Corr. 

r 

Corr. 

T 

Corr. 

T 

Corr. 

T 

Corr. 

6m 

o'.o 

i8m 

Is.  I 

30m 

5M 

42" 

I4M 

54m 

29s.  9 

7 

0  .1 

19 

I  -3 

31 

5  -7 

43 

15  .1 

55 

31-6 

8 

0  .1 

20 

I  -5 

32 

6  .2 

44 

16  .2 

56 

33  -3 

9 

0  .1 

21 

I  .8 

33 

6  .8 

45 

17  -3 

57 

35  -I 

10 

0  .2 

22 

2  .O 

34 

7  -5 

46 

18.5 

58 

37  -o 

ii 

O  .2 

23 

2-3 

35 

8  .2 

47 

19  .7 

59 

39  -o 

12 

0-3 

24 

2  .6 

36 

8  .9 

48 

21  .O 

60 

41  .0 

13 

0.4 

25 

3  -o 

37 

9.6 

49 

22  .3 

61 

43  -i 

14 

0.5 

26 

3  -3 

38 

10  .4 

50 

23  -7 

62 

45  -2 

15 

o  .6 

27 

3  -7 

39 

ii  -3 

51 

25-2 

63 

47  -4 

16 

o  .8 

28 

4  .2 

40 

12  .2 

52 

26.7 

64 

49  -7 

i? 

0.9 

29 

4.6 

41 

13  -I 

53 

28  .3 

65 

52  .1 

GEODETIC  ASTRONOMY. 


§307. 


307. 

2  sin«  \t 

(See  g  172.) 

sin  i" 

t 

OP 

im 

2m 

3m 

4m 

5m 

6m 

7m 

8m 

0» 

o".oo 

".96 

~^8s~ 

i  7".  67 

31".  42 

49".  09 

70".  68 

96".  20 

I25".  65 

I 

0  .00 

7  .98 

17  .87 

31  .68 

49  .41 

71  -07 

96  .66 

126  .17 

2 

o  .00 

.10 

8   .12 

18  .07 

3i  -94 

49  -74 

7i  -47 

97  '12 

126  .70 

3 

0  .00 

.16 

8  .25 

18  .27 

32  .20 

50  .07 

71  .86 

97  -58 

127  .22 

4 

0  .01 

•  23 

8  -39 

18  .47 

32  -47 

50  .40 

72  .26 

98  .04 

"7  -75 

5 

0  .01 

31 

8  .52 

i3  .67 

S2  -74 

50  .73 

72  .66 

98  .50 

128  .28 

6 

0  .02 

.38 

8  .66 

18  .87 

33  -oi 

5i  -07 

73  .06 

98  .97 

128  .8l 

7 

o  .02 

'45 

8  .80 

19  .07 

33  -27 

51  *4° 

73  -46 

99  -43 

129  .34 

8 
9 

o  .03 
o  .04 

:£ 

8  .94 
9  .08 

19  .28 
19  -48 

33  -54 
33  -8i 

5i  -74 
52  .07 

73  .86 
74  .26 

99  -90 
100  .37 

129  .87 
130  .40 

10 

n 

12 

o  .05 

o  .06 
o  .08 

.67 

9  .22 

9  .36 
9  -So 

19  .69 
19  .90 

20  .11 

34  -09 

34  .|6 
34  -64 

52  .41 

52  -75 
53  -09 

74  -66 

75  -06 
75  -47 

100  .84 
ioi  .31 

101   .78 

I30  .94 
131  -47 

132  .01 

j. 

o  .09 

.91 

9  -64 

20  .32 

34  -91 

53  -43 

75  -88 

IO2  .25 

132  .55 

14 

0  .11 

.99 

9  -79 

20  .53 

35  -19 

53  -77 

76  .29 

102  .  72 

133  .09 

j- 

0  .12 

3  -07 

9  -94 

20  .74 

35  -46 

54  -ii 

76  .69 

103  .20 

133  .63 

10 

o  .14 

3  '15 

10  .09 

20  .95 

35  -74 

54  -46 

77  .10 

103  .67 

*7 

o  .16 

3  '23 

10  .24 

21   .16 

36  .02 

54  -80 

77  -Si 

104  .15 

134  .71 

18 

o  .18 

3  -32 

10  .39 

21  .38 

36   .  T.O 

55  -15 

77  -93 

104  .63 

135  .25 

19 

o  .20 

3  -4° 

10  .54 

21   .60 

36   .58 

55  -So 

78  -34 

105  .10 

135  .80 

20 

0  .22 

3  '49 

10  .69 

21   .82 

36   .87 

55  .84 

78  -75 

105  .58 

136  .34 

21 

o  .24 

3  -S8 

10  .84 

22  .03 

37  .15 

56  .19 

79  -16 

106  .06 

136  .88 

22 

o  .26 

II  .00 

22   .25 

37  -44 

S6  -55 

79  -58 

106  .55 

*37  -43 

23 

o  .28 

3  -76 

II  .15 

22  .47 

37  -72 

56  .90 

80  .00 

107  .03 

137  .98 

24 

o  .31 

3  -85 

II  .31 

22   .70 

38  .01 

57  -25 

80  .42 

107  .51 

J38  .53 

25 

o  -34 

3  -94 

II  .47 

22   .92 

38  .30 

57  -60 

80  .84 

107  -99 

139  -08 

26 

11 

29 

o  .37 
o  .40 
o  .43 
o  .40 

4  -03 
4  .12 
4  .22 
4  .32 

II  .63 

II  .79 

ii  '95 

12  .11 

23  .14 

23  -I! 
23  .60 

23  .82 

38  -59 
38  .88 
39  .17 
39  .46 

57  -96 
58  .32 
58  .68 
59  -03 

81  .26 
81  .68 
82  .10 
82  .52 

108  .48 
108  .97 
109  .46 
109  .95 

139  -63 
140  .18 
140  .74 
141  -29 

3° 

o  .49 

4  -42 

12  .27 

24  .05 

39  -76 

59  .40 

82  -95 

no  .44 

141  -85 

3i 

o  .52 

12  .43 

24  .28 

40  .05 

59  -75 

83  -38 

no  .93 

142  .40 

33 

34 

o  .56 
o  -59 
o  .63 

4  .62 
4  .72 
4  .82 

12  .60 
12  .76 
12  -93 

24  .51 

24  -74 
24  -98 

40  -35 
40  .65 
40  -95 

60  .11 

60  .47 
60  .84 

83  .81 
84  -23 

84  .66 

in  .43 
in  .92 

112  .41 

142  .96 
M3  -52 
144  .08 

35 

o  .67 

4  -92 

13  .10 

25  .21 

4i  .25 

61  .20 

85  -09 

112  .90 

144  >64 

36 
37 

o  .71 
o  .75 

5  .°3 
5  'X3 

13  '27 
13  "44 

25  -45 
25  .68 

::  -M 

61  -57 
61  .94 

85  -52 

85  .95 

113  -40 

"3  .9° 

145  -20 
145  -76 

38 
39 

o  .79 
o  .83 

5  -24 
5  -34 

13  .62 

13  -79 

26  .  16 

42  .15 
42  .45 

62  .31 

62  .68 

86  .39 
86  .82 

114  .40 
114  .90 

146  -33 
I46  .89 

40 

41 

o  .87 
o  .91 

5  -45 

13  .96 
14  -13 

26  .40 
26  .64 

42  .76 

43  -06 

63  .05 

63  .42 

87  .26 
87  .70 

115  .40 

"5  -90 

147  -46 
148  .03 

42 

o  .96 

5  -67 

14  .31 

26  .88 

43  .37 

63  -79 

88  .14 

116  .40 

148  .60 

43 

I  .01 

5  -78 

14  .49 

27  .12 

43  -68 

64  .16 

88  .57 

116  .90 

149  '17 

44 

I  .06 

5  -90 

14  -67 

27  "37 

43  «99 

64  -54 

89  .01 

117  .41 

149  -74 

45 

I  .10 

6  .01 

14  .85 

27  .6l 

44  -30 

64  .91 

89  -45 

117  .92 

150  .31 

46 

i  -*5 

6  .13 

IS  -03 

27  .86 

44  '61 

65  -29 

89  .89 

118  %43 

150  .88 

47 

I  .20 

6  .24 

15   -21 

28  .10 

44  .92 

65  .67 

90  -33 

118  .94 

151  -45 

48 

I  .26 

6  .36 

IS  -39 

28  .35 

45  -24 

66  .05 

90  .78 

"9  -45 

152  .03 

49 

I  .31 

6  .48 

15  -57 

28  .60 

45  -55 

66  .43 

91  .23 

119  .96 

152  .61 

5° 

I  .36 

6  .60 

15  -76 

28  .85 

45  -87 

66  .81 

91  .68 

120  .47 

153  -19 

51 

I  .42 

6  .72 

15  -95 

29  .10 

46  .18 

67  .19 

92   .12 

120   .98 

153  -77 

52 

I  .48 

6  .84 

16  .14 

29  .36 

46  .50 

67  .58 

92   -57 

121  .49 

154  -35 

53 

1  -53 

6  .06 

16  .32 

29  .61 

46  .82 

67  .96 

93  .02 

122  .OI 

154  -93 

54 

i  -59 

7  -09 

16  .51 

29  .86 

47  .14 

68  .35 

93  -47 

122  .53 

»55  -Si 

55 

i  .65 

7  .21 

16  .70 

30  .12 

47  '46 

68  .73 

93  «92 

123  .03 

156  .09 

56 

i  .71 

7  «34 

16  .89 

30  .38 

47  -79 

69  .12 

94  -38 

123  .57 

156  -67 

57 

i  .77 

7  -46 

17  .08 

30  .64 

48  .n 

69  .5I 

94  -83 

T24  .09 

157  -35 

58 

i  .83 

7  .60 

17  .28 

30  .90 

48  .43 

69  .90 

95  -29 

124  .61 

1^7  -8.1 

59* 

i  .89 

7  -72 

17  -47 

31   .16 

48  .76 

70  .29 

95  -74 

125  -'3 

158  -43 

§307- 


TABLES. 


317 


_  2  sin2  \t 
sin  i" 


t 

9m 

10™ 

nm 

I2m 

I3» 

I4m 

i5m   |   16- 

0* 

159"  -02 

196".  32 

237"  -54 

282".  68 

33i".  74 

384".  74 

44i".  63 

502".  46 

I 

159  .6l 

196  .97 

238  .26 

283  -47 

S32  -59 

385  -65 

442  .62 

5»3  -5° 

2 

3 

160  .20 
160  .80 

197  .63 
198  .28 

238  .98 
239  -70 

284  .26 
285  .04 

333  -44 
334  -29 

3f6  .56 
387  .48 

443  .60 
444  -58 

5«4  -55 
505  -60 

4 

161  .39 

198  .94 

240  .42 

285  .83 

335  ^5 

388  .40 

445  -56 

506  .65 

5 

161  .98 

199  .60 

241  .14 

286  .62 

336  -oo 

389  -32 

446  -55 

507  .70 

6 

162  .58 

200  .26 

241  .87 

287  .41 

336  .86 

390  .24 

447  -54 

508  .76 

7 

163  .17 

200  .92 

242  .60 

288   .20 

337  -72 

39i  -16 

448  .53 

509  -81 

8 

163  .77 

201  .59 

243  -33 

289  .00 

338  .58 

392  .09 

449  -Si 

510  .86 

9 

164  -37 

202  .25 

244  .06 

289  .79 

339  -44 

393  -OI 

450  .50 

511  .92 

10 

164  .97 

202  .Q2 

244  -79 

290  .58 

34<>  -30 

393  -94 

451  .50 

5«  -98 

ii 

165  .57 

203  .58 

245  -52 

291   .38 

341  .16 

394  -86 

452  -49 

5M  -03 

12 

166  .17 

204  .25 

246  .25 

292   ,l8 

342  .02 

395  -79 

453  -48 

5'5  -09 

J3 

166  .77 

204  .92 

246  .98 

292   .98 

342  .88 

396  .72 

454  .48 

5*6  .15 

H 

167  -37 

205  .59 

247  -72 

293   -78 

343  -75 

397  -65 

455  .47 

517  .21 

15 

167  .97 

206  .26 

248  .45 

294  -58 

344  -62 

398  .58 

45<5  -47 

518  .27 

16 

168  .58 

206  .93 

249  «!9 

295   .38 

345  -49 

399  -52 

457  -47 

519  -34 

17 

169  .19 

207  .60 

249  -93 

346  -36 

400  .45 

458  -47 

520  .40 

18 

169  .80 

208  .27 

250  .67 

296  .99 

347  -23 

401  .38 

459  -47 

521  -47 

19 

170  .41 

208  .94 

251  .41 

297  -79 

348  .10 

402  .32 

460  .47 

522  .53 

20 

171   .02 

209  .62 

252  .15 

298  .60 

348  -97 

403  -26 

461  .47 

523  .60 

21 

171  .63 

2IO  .30 

252  .89 

299  .40 

349  -84 

404  .20 

462  .48 

524  -67 

22 
23 

172  .24 
172  .85 

210  .98 

211  .66 

253  -63 
254  -37 

300  .21 
3OI   .  O2 

350  -71 
35i  -58 

405  .14 
406  .08 

463  -48 
464  .48 

525  -74 
526  .81 

24 

J73  -47 

212  .34 

255  -12 

3°I   -S3 

35«  -46 

407  .02 

465  .49 

527  -89 

25 

174  .08 

213  .02 

255  -87 

302   .64 

353  -34 

407  .96 

466  .50 

528  .96 

26 

174  .70 

213  .70 

256  .62 

303   -46 

354  -22 

408  .90 

467  -Si 

53°  -03 

27 

175  -32 

2I4  .38 

257  -37 

3<>4  -27 

355  -"o 

409  .84 

468  .52 

531  •" 

28 

175  -94 

215  «°7 

258  .12 

305   .09 

410  .79 

469  -53 

532  .18 

29 

176  .56 

215  -75 

258  .87 

305   .90 

356  -86 

4"  -73 

470  -54 

533  -26 

3° 

177  .18 

216  .44 

259  .62 

306  .72 

357  -74 

412  .68 

47i  -55 

534  .33 

31 

177  .80 

217  .12 

260  .37 

307  -54 

358  .62 

4i3  -63 

472  -57 

535  -4i 

32 

178  -43 

217  .81 

26l   .12 

308  .36 

359  -Si 

4M  -59 

473  -58 

536  -5° 

33 

179  .05 

218  .50 

261  .88 

309  .^8 

360  -39 

4*5  -54 

474  .60 

537  -58 

34 

179  .68 

219  .19 

262  .64 

310  .00 

361  .28 

416  .49 

475  -62 

538  .67 

35 

180  .30 

219  .88 

263  .39 

310  .82 

362  .17 

4*7  -44 

476  .64 

539  -75 

36 

180  .93 

220  .58 

264  .15 

311  .65 

363  .07 

418  .40 

477  -65 

540  .83 

37 

181  .56 

221   .27 

264  .91 

312  -47 

363  .96 

4*9  -35 

478  -67 

54t  -9i 

38 

182  .19 

221   .97 

265  .68 

3i3  -30 

364  .85 

420  .31 

479  .70 

543  -oo 

39 

182  .82 

222  .66 

266  .44 

314  .12 

365  -75 

421  .27 

480  .72 

544  -°9 

40 

183  .46 

223  .36 

267  .20 

3H  -95 

366  .64 

422  .23 

481  .74 

545  -18 

4i 

184  .09 

224  .06 

267  .96 

3'5  -78 

367  -53 

423  -19 

482  .77 

546  .27 

42 

184  .72 

224  .76 

268  .73 

316  .61 

368  .  42 

424  -«5 

483  -79 

547  -36 

43 

185  -35 

225  .46 

269  .49 

317  -44 

3^9  -3i 

425  «" 

484  .82 

548  -45 

44 

185  .99 

226  .l6 

270  .26 

318  -27 

370  -21 

426  .07 

485  -85 

549  -55 

45 

186  .63 

226  .86 

271   .02 

319  '10 

37i  •" 

427  -°4 

486  .88 

55°  -64 

46 

187  .27 

227  .57 

271   .79 

319  '94 

372  .01 

428  .01 

487  .91 

551  -73 

47 

187  .91 

228  .27 

272   .56 

320  '78 

372  -91 

428  .97 

488  .94 

552  .83 

48 

188  .55 

228  .98 

273  -34 

321  .62 

373  -82 

429  -93 

489  .97 

553  -93 

49 

189  .19 

229  .68 

274  •" 

322  .45 

374  -72 

43°  -9° 

491  .01 

555  -03 

50 

189  .83 

230  .39 

274  .88 

323  -29 

375  -62 

431  -87 

492  .°5 

556  -13 

51 

190  .47 

231  .10 

275  -65 

324  -T3 

376  -52 

432  -84 

493  .08 

557  -24 

52 

I9I   .12 

231  .81 

276  -43 

324  -97 

377  -43 

433  -82 

494  .12 

558  -34 

53 

igi   .76 

232...  52 

277   .20 

325  -8i 

378  -34 

434  -79 

495  -i5 

559  -44 

54 

192   .41 

233  -.24 

277   -98 

326  .66 

379  .26 

435  -76 

496  .19 

56o  .55 

55 

193   .06 

233  -95 

278   .76 

327  -5° 

380  .17 

436  -73 

497  -23 

5°i  -65 

56 

193   .71 

234  -67 

279  -55 

328  .35 

381  .08 

437  -71 

498  .28 

562  .76 

57 

194   -36 

»35  -38 

280  .33 

329  -19 

381  .99 

438  -69 

499  -32 

563  -87 

58 
59 

195   '*A 

195  .66 

236  .10 
236  .82 

28l   .12 
28l   .90 

33°  -04 
330  .89 

382  .90 
383  -82 

439  -67 
440  .65 

5°o  -37 
501  .4x 

564  .98 
566  .08 

GEODETIC  ASTRONOMY. 


2  sin2  \t 
sin  i" 


17- 

,8- 

19- 

20m 

.,« 

22m 

23" 

24- 

25- 

o1 

567".  2 

~635".9~ 

70S".  4 

784".  9 

865".  3 

949".  6 

1037".  8 

1129".  9 

1225".  9 

I 

568  .3 

637  .0 

709  .7 

786   .2 

866  .6 

951  .0 

1039  .3 

1131  .4 

1227  .5 

2 

569  -4 

638  .2 

710  .9 

787  -5 

868  .0 

952  -4 

1040  .8 

"33  -o 

1229   .2 

3 

570  -5 

639  -4 

712  .1 

788  .8 

869  .4 

953  -8 

1042  .3 

"34  -6 

1230  .8 

4 

571  -6 

640  .6 

7^3  -4 

790  .1 

870  .8 

955  -3 

1043  .8 

1136  .2 

1232  .5 

s 

572  .8 

641  .7 

714  .6 

791  -4 

872  .1 

956  -7 

1045  .3 

"37  -8 

1234  .1 

6 

573  -9 

642  .9 

715  .9 

792  .7 

873  -5 

958  .2 

1046  .8 

"39  -3 

I235  -7 

7 

575  -° 

644  .1 

717  .1 

794  -o 

874  .9 

959  -6 

1048  .3 

1140  .9 

I237  .3 

8 

576  .1 

645  -3 

718  .4 

795  -4 

876  .3 

961  .1 

1049  -8 

"42  -5 

1239  .0 

9 

577  -2 

646  .5 

719  .6 

796  .7 

877  .6 

962  .5 

1051  .3 

1144  .0 

1240  .6 

10 

578  -4 

647  .7 

720  .9 

798  .0 

879  .0 

963  -9 

1052  .8 

H45  -6 

1242  .3 

ii 

579  -5 

648  .9 

722  .1 

799  -3 

880  .4 

965  .4 

I054  -3 

"47  -2 

1243  .9 

12 

580  .6 

650  .0 

723  -4 

800  .7 

881  .8 

966  .9 

I055  -9 

1148  .8 

1245  -6 

13 

581  -7 

651   .2 

724  -6 

802  .0 

883  .2 

968  .3 

1057  .4 

"So  .4 

1247  .2 

14 

582  .9 

652  .4 

725  -9 

803  -3 

884  .6 

969  .8 

1058  .9 

1152  .0 

1248  .9 

15 

584  -o 

653  -6 

727  .2 

804  .6 

886  .0 

971  .2 

1060  .4 

"53  -6 

1250  .5 

16 
17 

585  .1 

586  .2 

654  -8 
656  .0 

728  .4 
729  .7 

806  .0 
807  .3 

887  .4 
888  .8 

972  -7 
974  -i 

1062  .0 
1063  .5 

"55  .2 
1156  .8 

1252  .2 

1253  .8 

18 

587  -4 

657  -2 

730  -9 

808  .6 

890  .2 

975  -5 

1065  .0 

1158  .3 

1255  -5 

X9 

588  .5 

658  .4 

732  -2 

809  .9 

891  .6 

977  -o 

1066  .5 

"59  '9 

1257  .1 

20 

589  -6 

659  -6 

733  -5 

8n  .3 

893  .0 

978  -5 

1068  .1 

1161  .5 

1258  .8 

21 

590  .8 

660  .8 

734  -7 

812  .6 

804  -4 

979  -9 

1069  .  6 

"63  .1 

1260  .5 

22 

591  -9 

662  .0 

736  .0 

813  .9 

895  .8 

981  .4 

1071  .1 

1164  .7 

1262  .2 

23 

593  -° 

663  .2 

737  -3 

8i<;  .2 

897  .2 

982  .9 

1072  .6 

1166  .3 

1263  .8 

24 

594  -2 

664  .4 

738  -5 

816  .6 

898  .6 

984  .4 

1074  .2 

1167  .9 

1265  .5 

25 

595  -3 

665  .6 

739  -8 

817  .9 

900  .0 

985  .8 

1075   .7 

1169  .5 

1267  .  i 

26 

596  -5 

666  .8 

741  .1 

819  .2 

901  .4 

987  -3 

1077   .2 

1171  .1 

1268  .8 

27 

597  -6 

668  .0 

742  -3 

820  .5 

902  .8 

988  .8 

I078   .7 

1172  .7 

1270  .5 

28 

598  -7 

669  .2 

743  -6 

821  .9 

904  .2 

990  -3 

1080  .3 

"74  -3 

1272  ,i 

29 

599  -9 

670  .4 

744  -9 

823  .2 

905  -6 

991  .8 

1081  .8 

"75  -9 

1273  .7 

3° 

601  .0 

671  .6 

746  .2 

824  .6 

907  .0 

993  -2 

1083  -3 

"77  -5 

1275  -4 

31 

602  .2 

672  .8 

747  -4 

825  .9 

908  .4 

994  -7 

1084  .8 

"79  -1 

1277  .1 

32 
33 
34 

603  -3 
604  .5 
605  .6 

674  .1 
675  -3 
676  .5 

748  .7 
750  .0 
751  .3 

827  .3 
828  .6 
829  .9 

909  .8 

911   .2 

912  .6 

996  .2 
997  .6 
999  -i 

1086  .4 
1087  .9 
1089  .5 

1180  .7 
1182  .3 
"83  .9 

1278  .8 
1280  .4 
1282  .1 

35 

606  .8 

677  .7 

752  .6 

831  .2 

914  .0 

1000  .6 

1091  .0 

"85  -5 

1283  .8 

36 

607  .9 

678  .9 

753  -8 

832  .6 

9i5  -5 

1002  .1 

1092  .6 

1187  .1 

1285  .5 

11 

609  .1 
610  .2 

680  .1 
681  .3 

755  -i 
756  -4 

833  -9 

916  .9 
918  .3 

1003  .5 

1005  .0 

1094  .1 
1095  -7 

1188  .7 
"9°  -3 

1287  .1 
1288  .8 

39 

6n  .4 

682  .6 

757  -7 

836  .6 

919  .7 

1006  .5 

1097  .2 

1191  .9 

1290  .5 

40 

612  .5 

683  .8 

759  -o 

838  .0 

921  .1 

1008  .0 

1098  .8 

"93  -5 

1292  .3 

41 

613  -7 

685  .0 

760  .2 

839  -3 

922  .5 

1009  .4 

IIOO  .3 

"95  •* 

1293  .8 

42 

614  .8 

686  .2 

76l   -5 

840  .7 

923  -9 

1010  .9 

nor  .9 

1196  .7 

1295  -5 

43 

616  .0 

687  .4 

762  .8 

842  .0 

925  -3 

IOI2  .4 

1103  .4 

"98  .3 

1297   .2 

44 

617  .2 

688  .7 

764  .1 

843  -4 

926  .8 

1013  .9 

1105  .0 

"99  -9 

1298  .9 

45 

618  .3 

689  .9 

765  -4 

844  .7 

928  .2 

1015  .4 

1106  .5 

1201  .5 

1300  .5 

46 

619  .5 

691  .1 

766  .7 

846  .1 

929  .6 

1016  .9 

1108  .1 

1203  .1 

I  302   .  2 

47 

620  .6 

692  .4 

768  .0 

847  -5 

931  .0 

ioiS  .4 

1109  .6 

1204  .7 

1303   .9 

48 

621  .8 

693  -6 

769  -3 

848  .9 

932  .4 

1019  .9 

IIII  .2 

I206  .4 

1305  -6 

49 

623  .0 

694  .8 

770  .6 

850  .2 

933  -8 

IO2I  .4 

III2  .7 

I2O8  .O 

1307  -3 

50 

624  .1 

696  .0 

771  .9 

851  .6 

935  -2 
936  .6 

1022  .8 

III4  .3 

1209  .6 

1309  .0 

51 
52 

625  .3 

626  .5 

697  .3 
698  .5 

'773  -I 
774  -5 

854  0 

938  .1 

1025  .8 

III7  .4 

1212  .9 

1312  .4 

53 

627  .6 

699  .7 

775  -7 

855  -7 

939  -5 

1027  .3 

1118  .9 

1214  .5 

1314  .1 

54 

628  .8 

701  .0 

777  .1 

857  -I 

940  .9 

1028  .8 

1120  .5 

1216  .1 

1315  -7 

55 

630  .0 

7O2   .2 

778  .4 

858  .4 

942  -3 

1030  .  3 

1122  .O 

1217  .7 

1317  .4 

56 

631  .2 

703  -5 

779  -7 

859  .8 

943  .8 

1031  .8 

1123  .6 

1219  .4 

1319  .1 

57 

632  .3 

704  .7 

781  .0 

861  .1 

945  -2 

1033  .3 

1125  .1 

1221  .O 

1320  .8 

58 

633  -5 

705  -9 

782  .3 

862  .5 

946  .6 

1034  .8 

1126  .7 

1222  .6 

1322  .5 

59 

634  -7 

707  .1 

783  .6 

863  .9 

948  .1 

1036  .3 

1128  .3 

1224  .2 

1324  -2 

§307- 


TABLES. 


319 


2  sin* 


—  2  sin<  it 
sin  i" 


2  sin*  |/ 
sin  i" 


t 

26- 

27" 

28m 

29" 

OP 

1325". 

J429". 

I537"-- 

1649".  o 

I 

1327  • 

1431   . 

1539  .; 

1650  .9 

2 

1329  . 

1433   • 

1541  .: 

1652  . 

3 

1  33  1  . 

1434  • 

1654  .7 

4 

1332  . 

1436  . 

1544  .« 

1656  .6 

5 

1334  • 

1438  . 

1546  •€ 

1658  .5 

6 

1336  . 

1440  . 

1548  .4 

1660  .4 

7 

1337  • 

I442   . 

1550  .2 

1662  .3 

8 

H43  • 

1664  .2 

9 

1341  • 

M45  • 

1553  -9 

1666  .1 

10 

1342  . 

1447  • 

1555  -8 

1668  .0 

i 

1344  • 

1449  • 

1557  -6 

1669  .9 

2 

1346  - 

*559  -5 

1671   .( 

3 

1348  . 

1452  • 

1561  .'. 

1673  -8 

4 

1349  . 

1454  • 

1563  -2 

1675  -7 

1 

1353  • 

1456  . 
1458  . 

1565  .0 

I566  .9 

1677  .6 
l679  -5 

7 

1354  • 

1459  • 

1568  .7 

1681  .4 

8 

1461  . 

I57°  -5 

1683  .3 

9 

1358  '. 

1463  . 

1572  .4 

1685  .2 

20 

1360  . 

1465  . 

J574  -3 

1687  -2 

21 

1361  . 

1466  . 

1576  .1 

1689  .1 

22 

1363  -5 

1468  . 

1578  .0 

1691  .0 

23 

1365  .2 

1470  . 

1579  -8 

1692  .9 

24 

1367  .0 

1472  .3 

1581  .7 

1694  .8 

25 

1368  .7 

1474  . 

1583  -5 

1696  .7 

26 

1370  .4 

1475  -9 

1585  -3 

1698  .6 

27 

1372  . 

1477  -7 

1587  -2 

1700  .5 

28 

1373  -9 

T479  -5 

1589  -i 

1702  .5 

29 

1375  -6 

1481  .3 

1590  .9 

1704  .4 

30 

1377  -3 

1483  .1 

1592  .7 

1706  .3 

31 

1379  -0 

1484  .9 

1594  .6 

1708   .2 

32 

1380  .8 

1486  .7 

*596  -5 

1710  .2 

33 

1382  .5 

1488  .5 

1598  .3 

1712   .1 

34 

1384  .2 

H90  .3 

1600  .2 

1714  .0 

11 

1385  -9 
1387  -7 

1492  .1 

l6o2   .  I 

1604  .0 

1715   -9 

1717   .9 

37 

1389  .4 

T495  '7 

1605   .9 

1719  .8 

38 

1391   -2 

J497  -5 

1607  .7 

1721  .7 

39 

1392   .9 

1499  -3 

1609  .6 

1723  -6 

40 

1394  -7 

1501  .i 

1611  .5 

1725  .6 

41 

1396  .4 

1502  .9 

1613  .3 

1727  -5 

42 

1398  -2 

1504  -7 

I6l5   .2 

43 

X399  -9 

1506  .5 

1617  .1 

1731  •' 

44 

1401  .7 

1508  .4 

1619  .0 

1733  -4 

n 

1405   .2 

1510  .2 

1512  .0 

1620  .8 
1622  .7 

*735  -3 

1737  .2 

47 

1406  .9 

1513  -8 

1624  .6 

1739  -2 

48 

1408   .7 

1515  .6 

1626  .5 

1741   .2 

49 

1410  .4 

5»7  -4 

1628  .3 

1743   .1 

So 

1412  .2 

519  -2 

630  .2 

1745  -I 

51 

I4I3  -9 

521   .O 

632  .1 

1747  .0 

52 

1415  -7 

522  .9 

634  .0 

1749  -0 

53 
54 

1417  -4 
1419  -2 

524   -7 
526  .5 

635  -9 
637  '7 

1750  .9 

1752  .8 

55 

1420  .9 

528  .3 

639  -6 

1754  -8 

56 

1422  .7 

530   .2 

641  .5 

1756  .8 

57 

1424  .4 

S32  .0 

643  -3 

1758  -7 

58 
59 

1426  .2 
1427   .9 

533  .8 
535  -6 

645  .2 
647  .1 

I7^°  'I 
1762  .6 

t_ 

« 

t 

it 

om 

o".oc 

20™  0 

"•4g 

I   0 

0  .OC 

10 

2   0 

O  .OC 

20 

J 

3  o 

0  .00 

30 

•65 

4  o 

0  .00 

40 

5  o 

o  .0 

50 

.76 

6  o 

0  .0 

21   0 

82 

7  o 

o  .0 

IO 

A 

8  o 

0  .0 

20 

.< 

9  o 

o  .o< 

3° 

,j 

10   0 

o  .09 

40 

.0 

II   0 

0  .  I 

50 

.12 

12   0 

O  .1 

22   O 

.19 

10 

0  .2 

10 

.2< 

20 

0  .2 

2O 

t- 

30 

0  .2 

3° 

.'. 

40 

o  .2) 

40 

•4 

50 

0  .2 

5° 

6 

10 

O  .2 

IO 

'.69 

20 

o  .30 

20 

•77 

30 

o  .3 

30 

•85 

4° 

o  .3 

40 

•9; 

So 

o  -3 

5° 

.01 

10 

o  .3? 

IO 

3  •  IO 
3  -18 

20 

20 

3  -27 

30 

4O 

o  .41 

3° 

3  -36 

50 

0  -45 

50 

3  -55 

15  o 

o  .47 

25  o 

3  -64 

10 

o  .49 

10 

3  -74 

20 

o  .52 

20 

3  -34 

30 

o  .54 

30 

3  -94 

40 

o  .56 

40 

4  -05 

50 

0  -59 

5° 

4  '*5 

16  o 

o  .61 

26  o 

4  .26 

IO 

o  .64 

TO 

4  -37 

20 

o  .67 

20 

3° 

o  .69 

3° 

4  .60 

50 

0  -75 

5° 

4  -83 

10 

o  .81 

10 

5  .08 

20 

0  .84 

20 

5  -20 

3° 

o  .88 

30 

5  -33 

40 

o  .91 

40 

5  -46 

50 

o  -95 

5° 

5  .60 

18  o 

o  .98 

8  o 

5  -73 

IO 

I   .02 

10 

5  -87 

20 

I   .06 

20 

'  -OI 

3° 

.09 

30 

6  .15 

40 

•  11 

40 

•3° 

50 

•  iS 

5° 

•44 

19  o 

.22 

9  ° 

•59 

IO 

.26 

10 

•75 

20 

•3° 

20 

.90 

3° 

•35 

30 

.06 

40 

.40 

40 

.22 

50 

•44 

So 

.38 

20  o 

•49 

o  o 

•55 

t 

O 

I4"» 

o' 

'.000 

*5 

o 

.001 

16 

o 

.001 

17 

O 

.001 

18 

o 

.002 

19 

0 

.002 

20 

0 

.003 

21 

o 

.004 

22 

0 

.005 

23 

o 

.007 

24 

o 

.009 

11 

o 
o 

.Oil 

.014 

27 

0 

.017 

28 

o 

.021 

29 

o 

.026 

30 

o 

.032 

320 


GEODE  TIC  A  S  TRONOM  Y. 


§308. 


308. 


(See  §  175.) 


Hour- 
angle  be- 
fore or 
after 

The  Correction  to  be  applied  to  the  Latitude  of  the  station  to  obtain 
the  apparent  altitude  of  Polaris.    Computed  for  the  declination 
88°  46'  and  the  mean  refraction. 

Correc- 
tion for 
i'  in- 
crease 

upper 
Culmina- 

Latitude 

Latitude 

Latitude 

Latitude 

Latitude 

Latitude 

Latitude 

in  the 
declina- 

tion. 

3o° 

35° 

40° 

45° 

50° 

55° 

60° 

Polaris. 

oh  oom 

4-1°  is'-  6 

4-i°  i5'-3 

4-i°  15'-  1 

4-i°  H'-9 

+  1o  I4,  8 

4-i°  i4'-6 

4-i°  14'-  5 

—  I'.Q 

15 

4-i    15  -4 

4-i    15-2 

4-i    14  -9 

+  i    14.8 

4-1     14^6 

4-i    14  .4 

4-i    14  .3 

I    .0 

3° 

4-i    14-9 

4-i    14  .7 

4-i    14.5 

4-i    14-3 

4-i    M  .2 

4-i    14  -o 

4-1     13  -8 

—  I    .0 

45 
oo 

4-1       14   .2 

4-1    13  .0 

+  1     13  -9 
4-i     12  .8 

4-i    13  -7 
4/i    I2  .5 

£  -3:l 

+  «     '3-3 

+  1        12    .2 

4-1        I3    .2 
4-1        12    .0 

j-i     13  -o 

4-i    ii  .9 

—  i  .0 

—  I    .0 

15 

4-i     ii  .6 

4-i    ii  .3 

4-i    ii  .1 

-t-i    10.9 

4-x     io  .8 

4-i     io  .6 

4-i    io  .4 

—  o  .9 

30 

45 

00 

4-1    09  .9 
4-i    07  .9 

4  i    05  .6 

4-i     09  .6 
+  1     07  .6 

4-1    05  .3 

4-1    09  .4 
4-i    07  .3 
4-i    05  .0 

+  1       09   .2 

4-i    07  .2 

+  1     04  .8 

4-i    09  .0 
4-i    07  .0 
4-i     04  .6 

4-i     08  .8 

4-i     06  .8 

4-1    04  .4 

4-i     08  .6 
4-1     06  .6 

4-i   04  .2 

—o  .9 

-o  .9 
—  o  .8 

15 

4-i    03  .0 

+  1       02    .7 

4-i    02  .4 

+  1       02    .2 

4"!       O2    ,O 

4-i     oi  .8 

+  i     oi  .6 

—  o  .8 

3° 

4"  I       OO    .  1 

4-o    59  .8 

4-o    59  .5 

+  0     59  .3 

4-o     59  .1 

4-o    58  .9 

4-o   58  .7 

-0.8 

45 

4-o     57  .0 

4-0   56.7 

TO     56  .5 

4"O       56   .2 

4-o     56  .0 

4-0     55  -8 

4-o     55  -5 

—  o  .7 

3     »5 

3     30 

4"0    53  .7 
-j-o    50  .1 

+  o    46  .4 

4-o    53  .4 
+  o     49.8 
+  0     46  .0 

4-0     53  .1 
4-0     49  -5 

+  o     45  -7 

4-o     49  .2 

4-0     45  .5 

4-o    49  .0 
4-0     45  .2 

+0    48  .8 
4-o     45  .0 

-|-o     52  .  i 

4-0    48  .5 

4-o     44  .7 

0  '7 
—  o  .6 
—  o  .6 

3     45 

4-o   42  .4 

4o   42  .1 

+  o     41  .8 

4o    41  .5 

4-0     41  .3 

+  0     41  .0 

4-o    40  .7 

—  °  -5 

4    oo 

4-0   38  .3 

-t-o     38  .0 

+  o     37  -6 

+o     37  -4 

4-o     37  .1 

+o    36  .8 

4-0    36  .5 

—  o  .5 

4     15 

4o     34  .0 

4-0     33  -6 

4-o     33  .3 

4-o     33  .0 

4-o     32  .8 

4-o   32  .5 

4-0    32  .1 

—  o  .4 

4     3° 

4-o    29  .6 

4"O       29   .2 

+  o    28  .9 

-4-0     28  .5 

4-o     28  .3 

4-0     28  .0 

4-0     27  .6 

—  o  .4 

4     45 

4-o     25  .0 

4o     24  .6 

+  o     24  .3 

4-0     24  .0 

+0     23  .7 

4-o    23  .4 

4o     23  .0 

—  o  .3 

5     oo 
5     15 
5     3° 

4-0       20   .4 

+  0     15  .6 
+  o     io  .8 

+  0       20   .0 

j-o    15.3 

+  0       10    .4 

4o    19  .7 
4-o    14  .9 

4o     io  .  i 

4-o    19  .4 
+  0     14  .6 

4-0      00    .9 

4-o    19  .1 
j-o    14.3 

4-0    09  .6 

4-o     18  .8  4-o     18  .4 
4-0     14  .o4o     13  .6 
+0     09  .2  +o     08  .8 

—  0   .2 
O   .2 
0    .1 

5     45 

-j-o     06  .0 

4-0     05  .6 

4-0   05  .3 

4-o    05  .0 

4-o   04  .7 

4-o    04  .4 

4-o     04  .0 

o  .0 

6    oo 

+  0      01    .2 

4"0     oo  .8 

+  o     oo  .5 

4"O      OO   .2 

—  o    oo  ,i 

—  o    oo  .5 

—  o     oo  .9 

0   .0 

6     15 

—  o    03  .6 

—  o     04  .0 

—  o     04  .4 

—  o     04  .7 

—  o    05  .0 

—  o    05  .3 

—  o     05  .7 

4-0  .1 

6    30 

—  o     08  .4 

—  o    08  .8 

0      09    .2 

—  o     09  .5 

—  0       09    .2 

—  o     io  .1 

—  o     io  .4 

-j-o.i 

6     45 
7     oo 

—  o     17  .9 

—  o     18  .3 

—  o     18  .6 

—  o     18  .9 

—  0       19   .2 

—  o    19  .6 

—  o     19  .9 

4-0.3 

7     15 

O      22   .5 

—  O      22    .9 

—  0      23    .2 

—  o    23  .6 

—  o    23  .8 

0       24    .2 

—  o     24  .6 

4-0.4 

7     3° 

—  o     27  .0 

—  o     27  .4 

—  o     27  .7 

—  o    28  .0 

—  o     28  .3 

—  o     28  .6 

—  o     29  .0 

4-0.4 

7     45 

—  o     31  .4 

—  0      31    .8 

—  o     32  .1 

—  o     32  .4 

—  o     32  .7 

—  o    33  .0 

—  o     33  -3 

4-0.5 

8     oo 

—  o    35  .6 

—  o    36  .0 

—  o    36  .3 

-o    36  .6 

—  o    36  .9 

—  o     37  .2 

—  o     37  -5 

4-o.s 

8     15 

—  o     39  '7 

—  o    40  .  i 

—  o    40  .4 

—  o    40  .7 

—  o     41  .0 

—  O       41    .2 

—  o     41  .6 

+  o  .6 

8     30 

—  o    43  .6 

—  o     44  .0 

—  o     44  -3 

—  o    44  .6 

-o    44  .8 

—  o     45-i 

—  o     45  .4 

4-o  .6 

8     45 

—  o     47  -3 

—  o     47  .7 

—  o     48  .0 

—  o     48  .3 

-o    48  .5 

—  o     48  .8 

—  o     49  .0 

4-0.7 

9    oo 

—  o     50  .8 

O       51    .  2 

—  o     5l  -5 

-o     51  .7 

—  o    51  .9 

—  o     52  .1 

—  o     52  .4 

-j-o  .7 

9     15 

—  o     54  .1 

—  °     54  -5 

—  o     54  -7 

—  o    55  -o 

—  o     55  -2 

—  °    55  -5 

—  o     55  -7 

4-0  .8 

9     30 

—  o     57  .2 

—  °     57  -5 

-o     57  -8 

—  o    58  .0 

—  o     58  .2 

-o     58  -5 

-o     58  .7 

4-o  .8 

9     45 

I       00   .0 

—  i     oo  .3 

—  i     oo  .  6 

—  i     oo  .8 

—  I      OI    .O 

—  I       01    .2 

—  i     oi  .4 

-t-o  .8 

10      OO 

—  i     02  .5 

—  i     02  .8 

—  i     03  .  i 

—  i     03  .3 

—  I   03  .4 

—  i     03  .6 

—  i     03  .9 

4-0  .9 

10    15 

—  i     04  .7 

—  i     05  .0 

—  i     05  .3 

—  i     05.5 

—  I    05  .7 

—  I    05  .9 

—  i     06  .1 

4-o  .9 

10    30 

—  i     06  .7 

—  1     07  .0 

—  I      07   .  2 

—  i     07  .5 

—  i    07  .6 

—I   07  .9 

—  i     08  .0 

4-o  .9 

io    45 

-i     08.4 

-I  08.7 

-I       08.9 

—  I      09    .2 

—  i    09  .3 

—I   09  .5 

—  I    09  .7 

4-o  .9 

II      00 

—  i    09  .8 

—  I       IO   .  I 

—  i     io  .5 

—  i     io  .6 

—  I       10  .9 

—  I        II    .0 

4-i  -o 

ii     15 

—  i     10  .8 

—  I     II  .1 

-i     ii  .4 

—  i     ii  .6 

-i     ii  .8 

—  I       12    .0 

1        12    .1 

4-1  .0 

ii     30 

—  i     ii  .6 

—I  II  .9 

—  I       12    .2 

—  I       12    .4 

-i     12.5 

—  I       12   .8 

1       12    .9 

4-1  -o 

"     45 

—I        12    .1 

-I   12.4 

—  I       12    .6 

—  I       12   .9 

—  i     13  .0 

—  I       I3   .2 

—  i     13  -3 

4-i  .0 

12      00 

—  i     12.3 

—  I        12    .6 

—  I        12    .8 

—  i     13  -o 

-I       I3    .2—1        13    .3 

-i     13-4 

4-i  .0 

§309- 


TABLES. 


321 


309.     CORRECTION    FOR   ERROR    OF   RUN  OF  A  MICROMETER 

The  tabular  value  is  the  correction  to  the  forward  reading. 

The  sign  of  the  correction  is  the  same  as  that  of  the  difference  Backward — Forward. 
Use  this  table  with  such  a  micrometer  as  is  described  in  §  189,  and  no  other. 


Forw'd 
Read- 

] 

3.  -  F 

Forw'c 
Read- 

ing. 

_//  _ 

_//  _ 

ing. 

2  .O 

3-o 

o'  15" 

o".o 

o".o 

o".o 

o".i 

o".i 

o".i 

o".i 

o".i 

0".2 

o'  15" 

o  30 

o".o 

0  .1 

0  .1 

0  .1 

0  .1 

0  .1 

0  .2 

O  .2 

0  .2 

o  -3 

o  30 

o  45 

0  .1 

O  .  I 

0  .1 

0  .2 

0  .2 

0  .2 

0  .2 

o  -3 

o  -3 

o  .4 

0  45 

I  OO 

O  .  I 

0  .1 

O  .2 

O  .2 

O  .2 

o  -3 

o  -3 

o  .4 

o  .4 

0  .6 

I  00 

I  15 

o".o 

O  .  I 

0  .2 

0  .2 

O  .2 

o  -3 

o  .4 

o  .4 

o  .4 

o  .5 

o  .8 

I  15 

I  30 

0  . 

0  .1 

0  .2 

0  .2 

o  -3 

o  .4 

o  .4 

o  -5 

o  -5 

o  .6 

o  .9 

I  30 

i  45 

o  . 

0  .1 

O  .2 

o  -3 

o  .4 

o  .4 

o  -5 

o  .6 

o  .6 

o  .7 

I  .0 

i  45 

2  00 

o  . 

O  .2 

0  .2 

o  -3 

o  .4 

o  -5 

o  .6 

o  .6 

o  .7 

o  .8 

.2 

2  OO 

2  15 

0  . 

O  .2 

o  -3 

o  .4 

o  .4 

o  -5 

o  .6 

o  .7 

o  .8 

o  .9 

•4 

2  15 

2  30 

0  . 

0  .2 

o  -3 

o  .4 

o  -5 

o  .6 

o  .7 

o  .8 

o  .9 

i  .0 

•5 

2  30 

2  45 

o  . 

0  .2 

o  -3 

o  .4 

o  .6 

o  .7 

o  .8 

o  .9 

.0 

.i 

.6 

2  45 

3  oo 

o  . 

O  .2 

o  .4 

o  -5 

o  .6 

o  .7 

o  .8 

.0 

.1 

.2 

.8 

3  oo 

3  15 

0  . 

o  -3 

o  .4 

o  -5 

o  .6 

o  .8 

o  .9 

.0 

.2 

•3 

2  .0 

3  15 

3  30 

0  . 

o  -3 

o  .4 

o  .6 

o  .7 

o  .8 

.0 

.1 

•3 

•4 

2  .1 

3  30 

3  45 

0  .2 

o  -3 

o  .4 

o  .6 

o  .8 

o  .9 

.0 

.2 

•4 

•5 

2  .2 

3  45 

4  oo 

O  .2 

o  -3 

o  -5 

o  .6 

o  .8 

I  .0 

.1 

•3 

•4 

.6 

2  .4 

4  oo 

4  15 

O  .2 

o  -3 

o  .5 

o  .7 

o  .8 

I  .0 

.2 

•4 

•  5 

•  7 

2  .6 

4  15 

4  30 

0  .2 

o  .4 

o  -5 

o  .7 

o  .9 

i  .1 

•3 

•4 

.6 

.8 

2  -7 

4  30 

4  45 

0  .2 

o  .4 

o  .6 

o  .8 

I  .0 

i  .1 

•3 

I  -5 

•7 

•9 

2  .8 

4  45 

5  oo 

O  .2 

o  .4 

o  .6 

o  .8 

I  .0 

I  .2 

•4 

I  .6 

.8 

2  .0 

3  -o 

5  oo 

322 


GEODETIC  ASTRONOMY. 


§310. 


310. 


Correction  for 

Hour-angle 
before  or  after 
Upper 

AZIMUTH  OF  POLARIS  COMPUTED  FOR 
DECLINATION  88°  46'. 

i'  Increase  in 
Declination 
of  Polaris. 

Culmination. 

Lat.  30°. 

Lat.  31°. 

Lat.  32°. 

Lat.  33°. 

Lat.  34°. 

Lat.  35°. 

Lat.  30°. 

oh  15™ 

o°  05'  40" 

o°  05'  43" 

o°  05'  47" 

o°  05'  51" 

o°  05'  55" 

o°  06'  oo" 

—  s" 

o  30 

o  ii  18 

o  n  25 

o  ii  33 

o  ii  41 

o  ii  49 

0   II   58 

—  9 

o  45 

I   00 

o  10  53 

0   22  23 

0   22   38 

o  22  53 

o  23  09 

o  23  26 

°  *7  53 
o  23  44 

—  18 

i  15 

o  27  48 

o  28  06 

o  28  25 

o  28  45 

o  29  06 

o  29  28 

—  23 

i  30 

o  33  05 

o  33  26 

o  33  49 

o  34  13 

o  34  38 

o  35  04 

—  27 

i  45 

o  38  13 

o  38  38 

o  39  04 

o  39  32 

o  40  oo 

o  40  3° 

—  31 

2   00 

o  43  12 

o  43  40 

o  44  09 

o  44  40 

o  45  12 

o  45  46 

—  35 

2   15 

o  47  58 

o  48  29 

o  49  02 

o  49  36 

0   50  12 

o  50  50 

—  39 

2   30 

o  52  32 

o  53  06 

o  53  42 

o  54  19 

0  54  59 

0  55  4° 

—  43 

2   45 

o  56  52 

o  57  29 

o  58  07 

o  58  48 

o  59  30 

i  oo  15 

—  46 

3  oo 
3  15 

I   00  58 

i  04  47 

i  oi  37 
i  05  28 

I   02   l8 
I   06   12 

i  03  oi 
i  06  58 

i  03  46 
i  07  46 

1  °4  34 

i  08  36 

—  50 

3  3° 
3  45 

i  "  33 

I   12   18 

I   09  48 
I   13  06 

i  13  56 

i  14  49 

i  15  45 

-1 

4  oo 

i  14  28 

!   I5   I5 

i  16  05 

i  16  57 

i  17  52 

i  18  50 

—  61 

4  15 

i  17  04 

I   I7   52 

i  18  44 

*  19  37 

i  20  34 

i  21  34 

-63 

4  30 

i  19  19 

I   2O  09 

I   21   02 

i  21  57 

i  22  55 

1  23  £7 

—  64 

4  45 

I   21   I4 

I   22   05 

i  22  59 

i  23  55 

i  24  55 

i  25  57 

—  66 

5  oo 

I   22  48 

I   23   40 

1  24  35 

i  25  32 

i  26  32 

i  27  36 

—  68 

5  15 

i  24  oo 

i  24  53 

i  25  48 

i  26  46 

i  27  47 

i  28  51 

-69 

5  30 

1   24  51 

i  25  44 

i  26  40 

i  27  38 

i  28  39 

i  29  44 

-69 

5  45 

I   25   20 

i  26  13 

i  27  09 

i  28  07 

t  29  09 

i  30  14 

—  70 

6  oo 

I   25  27 

i  26  19 

i  27  15 

i  28  14 

i  29  15 

i  30  20 

—  70 

6  15 

I   25   12 

i  26  04 

i  26  59 

i  27  57 

t  28  59 

i  30  03 

-69 

6  30 
6  45 

i  24  34 

i  23  36 

i  25  27 
i  24  27 

I   26  21 
I   25   21 

i  27  19 

i  26  18 

i  28  19 
i  27  17 

i  29  23 
i  28  20 

—  68 
—  67 

7  oo 

I   22   l6 

i  23  06 

i  23  59 

1  24  55 

i  25  53 

i  26  55 

—  66 

7  X5 

i  20  35 

I   21   25 

I   22   l6 

i  23  10 

I  24  08 

i  25  08 

-65 

7  30 

i  18  34 

I   19   22 

I   20  12 

I   21  05 

I   22  OO 

i  22  59 

—  64 

7  45 

i  16  13 

i  16  59 

I   I7   48 

i  18  39 

i  19  33 

i  20  29 

—  62 

8  oo 

i  13  33 

i  14  17 

1   J5  °4 

i  ^5  53 

i  16  45 

i  17  39 

—  60 

8  15 
8  70 

i  10  34 

i  ii  16 

I   12  01 

I   12   48 

i  *3  37 

i  14  29 

—  57 

8  45 

I    O7    17 

i  03  43 

i  07  57 

I   04   22 

I   05  02 

i  05  44 

i  06  29 

i  07  15 

54 
—  51 

9  oo 

o  59  54 

i  oo  30 

i  oi  07 

i  oi  47 

I   02  29 

I   03  12 

-48 

9  IS 

0  55  49 

o  56  23 

o  56  58 

o  57  34 

o  58  13 

o  58  54 

—  45 

9  30 
9  45 

o  51  31 
o  46  59 

o  52  oi 
o  47  27 

0  S2  34 
o  47  57 

o  53  08 
o  48  28 

o  53  43 
o  49  oo 

o  54  21 
o  49  34 

—  42 
-38 

10   00 

o  42  16 

o  42  42 

o  43  08 

o  43  36 

o  44  05 

o  44  35 

—  34 

10  15 

o  37  23 

o  37  45 

o  38  08 

o  38  33 

°  38  59 

o  39  26 

—  3° 

10  30 

0   32  20 

o  32  39 

0  32  59 

o  33  20 

o  33  43 

o  34  06 

—  26 

10  45 

o  27  09 

o  27  25 

o  27  42 

o  28  oo 

o  28  18 

o  28  38 

22 

II   OO 

O   21   51 

O   22   04 

0   22   l8 

0   22  32 

o  22  47 

o  23  03 

—  T8 

"  15 

o  16  28 

o  16  38 

o  16  48 

o  16  59 

o  17  10 

O   17   22 

—  13 

II  30 

0   II   01 

0   II   08 

o  ii  14 

O   II   22 

o  ii  29 

o  ii  37 

—  9 

"  45 
Elongation: 
Azimuth.... 

o  05  31 

I  25  27 

o  05  34 

I   26  20 

o  05  38 
i  27  16 

o  05  42 

I   28   14 

o  05  45 
i  29  16 

o  05  49 

I   30   20 

—  4 
-69 

h.  nt.  s 

h.  nt.  s. 

h.  nt.  s. 

h.  nt.  s. 

h.  nt.  s. 

h.  nt.  s. 

s. 

Hour-angle. 

5  57  09 

5  57  02 

5  56  55 

5  56  48 

5  56  40 

5  56  33 

~\~   2 

TABLES. 


323 


Hour-angle 
before  or  after 
Upper 
Culmination. 

AZIMUTH  OF  POLARIS  COMPUTED  FOR 
DECLINATION  88°  46'. 

Correction  for 
i'  Increase  in 
Declination 
of  Polaris. 

Lat.  35°. 

Lat.  36°. 

Lat.  37°. 

Lat.  38°. 

Lat.  39°. 

Lat.  40°. 

Lat.  40°. 

Oh  TSm 

o°  06'  oo" 

o°  06'  05" 

o°  06'  10" 

o°  06'  15" 

o°  06'  20"  ,0°  06'  26" 

—  5" 

o  30 

0   II   58 

0   12  08 

0   12   l8 

0   12   28 

o  12  39  o  12  50 

10 

o  45 

o  17  53 

o  18  07 

0   l8   22 

o  18  38 

o  18  54  o  19  ii 

—  16 

i  i5 

o  23  44 
o  29  28 

o  29  51 

o  30  15 

o  3°  41 

o  25  04  o  25  27 

o  31  08  p  31  36 

—  26 

i  30 

o  35  04 

o  35  3i 
o  41  02 

o  36  oo 
o  41  35 

o  36  31 

o  37  02  o  37  36 

—  31 
,6 

2   OO 

o  40  30 
o  45  46 

O   46  22 

o  47  oo 

o  47  39 

o  48  21  o  49  04 

3° 
—  40 

2   15 

o  50  50 

o  51  29 

o  52  ii 

o  52  55 

o  53  4*  o  54  29 

—  45 

2   30 

o  55  40 

o  56  23 

o  57  09 

o  57  57 

o  58  47  p  59  4° 

—  49 

2   45 

i  oo  15 

I   01   02 

i  oi  51 

i  02  43 

1  °3  37 

1  °4  34 

—  53 

3  15 

i  04  34 
i  08  36 

I   05   24 

I   09   29 

i  06  17 
i  10  25 

I   07   12 
I   II   24 

i  08  10 
i  12  25 

i  09  12 
i  13  30 

-S, 

fll 

3  3° 

3  45 

i  IS  45 

i  16  43 

i  14  14 
i  17  44 

i  18  49 

i  19  57 

I   21   08 

—  °3 
—  66 

4  oo 

i  18  50 

i  19  50 

i  20  54 

I   22  OI 

i  23  ii 

I   24   25 

-69 

4  15 

i  21  34 

I   22   36 

i  23  42 

I   24  51 

i  26  03 

I   27  20 

—  72 

4  30 

i  23  57 

i  25  oi 

i  26  08 

I   27   19 

i  28  33 

I   29  52 

—  74 

4  45 

i  25  57 

I   27   03 

I   28   12 

I   29  24 

i  30  40 

i  32  oo 

—  75 

5  oo 

i  27  36 

I   28   42 

I   29   52 

I   31  06 

i  32  23 

i  33  44 

—  76 

5  T5 

i  28  51 

i  29  59 

I   31   09 

I   32  24 

*  33  42 

i  35  °4 

—  77 

5  3° 

i  29  44 

i  30  52 

I   32   03 

i  33  18 

i  34  37 

*  35  59 

-78 

5  45 

i  3<>  14 

I   31   21 

i  32  33 

i  33  48 

i  35  07 

i  36  30 

-78 

6  oo 

I   30  20 

i  31  27 

i  32  39 

i  33  54 

i  35  13 

i  36  35 

-78 

6  15 

I   30  03 

i  31  10 

i  32  21 

i  33  36 

i  34  54 

i  36  16 

-78 

6  30 

I   29   23 

i  30  30 

i  31  40 

i  32  54 

i  34  " 

i  35  32 

—  77 

6  45 

I   28   2O 

i  29  26 

i  3°  35 

i  3i  48 

t  33  04 

1  34  24 

—  76 

7  oo 

i  26  55 

i  27  59 

i  29  07 

i  30  18 

»  3i  33 

i  32  52 

—  75 

7  15 

i  25  08 

i  26  ii 

i  27  17 

i  28  26 

i  29  39 

i  30  56 

—  73 

7  30 

i  22  59 

i  24  oo 

i  25  04 

I   26  12 

i  27  23 

i  28  38 

—  72 

7  45 

I   20   29 

I   21   28 

I   22   30 

I   23  36 

i  24  45 

i  25  57 

-69 

8  oo 

1  17  39 

i  18  36 

i  19  36 

i  20  39 

i  21  45 

i  22  54 

—  66 

8  15 

i  14  29 

i  15  24 

I   16  21 

I   17  22 

i  18  25 

i  19  31 

—  64 

8  30 

I   II   01 

i  "  53 

I   12   48 

'  13  45 

i  14  45 

i  15  48 

—  61 

8  45 

i  07  15 

i  08  04 

I  08  56 

i  09  50 

i  10  47 

i  ii  47 

-58 

9  oo 

I   03   12 

i  03  58 

i  04  47 

i  05  38 

i  06  31 

i  07  27 

—  54 

9  *5 

o  58  54 

o  59  37 

I   OO   22 

i  oi  09 

i  oi  59 

I   02   51 

—  5° 

9  3° 

o  54  21 

o  55  oo 

o  55  42 

o  56  25 

o  57  " 

o  57  59 

-46 

9  45 

0  49  34 

o  50  10 

o  50  48 

o  51  27 

o  52  09 

o  52  53 

—  42 

IO   OO 

o  44  35 

o  45  08 

o  45  42 

o  46  17 

o  46  54 

o  47  34 

-38 

10  15 

o  39  26 

0  39  54 

o  40  24 

o  40  55 

o  41  28 

o  42  03 

—  34 

10  30 

o  34  06 

o  34  30 

o  34  57 

o  35  24 

o  35  52 

O   36  22 

—  29 

10  45 

o  28  38 

o  28  59 

0   29  20 

o  29  43 

o  30  07 

o  30  32 

—  24 

II   00 

o  23  03 

o  23  19 

o  23  37 

o  23  55 

o  24  14 

o  24  35 

20 

11  !5 

0   17   22 

o  17  35 

o  17  48 

o  1  8  02 

o  18  16 

o  18  31 

—  15 

II  30 

o  ii  37 

o  ii  46 

0  "  54 

O   12   04 

0   12   13 

0   12  23 

—  10 

Elongation: 
Azimuth.... 

o  05  49 

I   30   20 

o  05  53 
i  31  28 

o  05  58 
i  32  40 

o  06  02 

»  33  55 

o  06  07 
i  35  M 

0   06  12 

I   36  36 

—  5 
-78 

h.  nt.  s. 

h,  nt.  s. 

h.  nt.  s. 

h.  nt.  s. 

h.  nt.  s. 

h.  nt.  t. 

s. 

Hour-angle. 

5  56  33 

5  56  25 

5  56  17 

5  56  09 

5  56  oo 

5  55  52 

+  3 

324 


GEODETIC  ASTRONOMY. 


§310. 


Hour-angle 
before  or  after 
Upper 
Culmination. 

AZIMUTH  OF  POLARIS  COMPUTED  FOR 
DECLINATION  88°  46'. 

Correctionfor 
i'  Increase  in 
Declination 
of  Polaris. 

Lat.  40° 

Lat.  41°. 

Lat.  42°. 

Lat.  43°. 

Lat.  44° 

Lat.  45° 

Lat.  40°. 

Oh  I5m 

o°  06'  26' 

o°  06'  32" 

0  06'  39" 

o°  06'  45' 

o°  06'  52' 

o°  07'  oo" 

~~~  5 

o  30 

0   12  50 

o  13  03 

13  is 

o  13  29 

o  13  43 

o  13  58 

10 

o  45 

0   I9   II 

o  19  30 

19  48 

o  20  08 

o  20  29 

0   20   52 

—  16 

1   00 

o  25  27 

o  25  51 

26  16 

o  26  43 

o  27  io 

o  27  40 

21 

I  15 

o  31  36 

o  32  05 

32  36 

o  33  09 

0  33  44 

o  34  21 

—  26 

I  30 

37  36 

o  38  ii 

38  48 

o  39  27 

o  40  09 

o  40  52 

—  31 

i  45 

43  26 

o  44  07 

44  50 

o  45  35 

o  46  22 

o  47  12 

-36 

a  oo 

49  04 

o  49  50 

50  39 

o  51  29 

o  52  23 

0  53  19 

—  40 

2   15 

54  29 

o  55  20 

56  14 

o  57  10 

o  58  io 

o  59  12 

—  45 

2   30 

59  40 

i  oo  35 

oi  34 

I   02   36 

I  03  41 

i  04  49 

—  49 

2   45 

°4  34 

i  05  34 

06  38 

i  07  44 

i  08  54 

I   10  08 

—  53 

3  oo 

09   12 

i  io  16 

ii  24 

1  12  35 

i  13  50 

i  15  09 

—  57 

3  15 

I   I3   30 

i  14  38 

15  So 

i  17  06 

i  18  25 

i  19  49 

-60 

3  3° 

I   17   29 

i  18  41 

»9  57 

I   21   l6 

i  22  39 

i  24  08 

—  63 

3  45 

I   21  08 

I   22  23 

i  23  42 

I  25  04 

I  26  32 

i  28  04 

—  66 

4  oo 

I   24  25 

i  25  43 

i  27  05 

i  28  31 

i  30  oi 

I  3i  37 

-69 

4  15 

I   27  2O 

i  28  40 

i  3°  °4 

1  3i  33 

i  33  07 

i  34  45 

—  72 

4  30 

I   29  S2 

i  3r  14 

i  32  41 

i  34  12 

i  35  48 

i  37  29 

—  74 

4  45 

i  32  oo 

i  33  24 

i  34  53 

t  36  25 

i  38  04 

i  39  47 

—  75 

5  oo 

i  33  44 

1  35  I0 

i  36  40 

i  38  14 

1  39  54 

i  4i  38 

-76 

5  i5 

1  35  04 

i  36  30 

I   38   02 

i  39  37 

i  41  18 

i  43  04 

—  77 

5  30 

i  35  59 

i  37  26 

I   38   58 

i  40  34 

i  42  16 

i  44  02 

-78 

5  45 

i  36  30 

i  37  57 

i  39  29 

i  41  05 

i  42  47 

i  44  34 

-78 

6  oo 

1  36  35 

I   38   02 

1  39  34 

i  41  io 

i  42  51 

i  44  S8 

-78 

6  15 

i  36  16 

i  37  43 

i  39  M 

i  40  49 

i  42  30 

i  44  16 

-78 

6  30 

i  35  32 

i  36  58 

i  38  28 

i  40  03 

i  41  42 

i  43  27 

—  77 

6  45 

i  34  24 

i  35  48 

i  37  17 

i  38  50 

i  40  28 

i  42  12 

-76 

7  oo 

i  32  52 

i  34  IS 

i  35  42 

*  37  13 

i  38  49 

i  40  31 

—  75 

7  i5 

i  30  56 

i  32  17 

i  33  42 

i  35  " 

i  36  45 

i  38  24 

—  73 

7  3° 

28  38 

i  29  56 

i  31  19 

i  32  46 

i  34  i7 

i  35  53 

—  72 

7  45 

i  25  57 

i  27  13 

i  28  33 

i  29  56 

i  3i  25 

i  32  58 

-69 

8  oo 

22  54 

i  24  07 

i  25  24 

i  26  45 

I   28   10 

i  29  40 

—  66 

8  15 

19  31 

i  20  41 

i  21  55 

i  23  12 

'  24  33 

i  25  59 

—  64 

8  30 

15  48 

i  16  55 

i  18  05 

i  19  18 

i  20  35 

i  21  57 

—  61 

8  45 

ii  47 

i  12  49 

i  13  55 

i  i5  05 

i  16  18 

i  17  35 

-58 

9  oo 

07  27 

i  08  26 

t  09  28 

i  io  33 

i  ii  41 

i  12  54 

—  54 

9  15 

i  02  51 

i  °3  45 

i  04  43 

i  05  43 

i  06  47 

i  °7  54 

—  50 

9  3° 

o  57  59 

o  58  49 

o  59  42 

I   00  38 

i  oi  37 

I   02   38 

-46 

9  45 

f>  52  53 

o  53  39 

o  54  27 

o  55  18 

o  56  ii 

o  57  07 

—  42 

io  oo 

0  47  34 

o  48  15 

o  48  58 

o  49  44 

o  50  32 

o  51  22 

-38 

10  15 
io  30 

o  42  03 

0   36   22 

o  42  39 
o  36  53 

o  43  18 

o  37  26 

o  43  58 
o  38  or 

o  44  40 

o  38  38 

o  45  25 

o  39  16 

—  34 
—  29 

1°  45 

o  30  32 

o  30  58 

o  31  26 

0  31  55 

o  32  26 

o  32  58 

—  24 

It   00 

o  24  35 

o  24  56 

o  25  18 

o  25  42 

o  26  06 

o  26  32 

—  20 

IT   15 

o  18  31 

o  18  47 

o  19  04 

0   19  22 

o  19  40 

o  20  oo 

—  IS 

II   30 

0   12  23 

o  12  34 

o  12  45 

o  12  57 

o  13  09 

o  13  23 

—  IO 

ii  45 

O   06   12 

o  06  18 

o  06  23 

o  06  29 

o  06  36 

o  06  42 

—  5 

Elongation: 
Azimuth.  ..  . 

I   36  36 

i  38  03 

1  39  35 

I  41  II 

42  53 

44  40 

-78 

h  m  s 

h.  tn  s 

t.  W   J" 

ft  ttl   S 

'.  fft   S 

t  .  ftt  ,  S  . 

J 

Hour-angle. 

5  55  52 

5  55  43 

5  55  34 

5  55  24 

55  M 

55  °4 

+  3 

TABLES. 


325 


Hour-angle 
before  or  after 
Upper 
Culmination. 

AZIMUTH  OF  POLARIS  COMPUTED  FOR 
DECLINATION  88°  46'. 

Correction  for 
i'  Increase  in 
Declination 
of  Polaris. 

Lat.  45°. 

Lat.  46°. 

Lat.  47°. 

Lat.  48°. 

Lat.  49°. 

Lat.  50°. 

Lat.  50°. 

o*  i5m 

o°  07'  oo" 

o°  07'  08" 

o°  07'  16' 

o°  07'  25" 

0  07'  34' 

o°  07'  44' 

—  6" 

3<> 

o  13  58 

o  14  13 

o  14  30 

o  14  48 

15  06 

o  15  25 

—  T3 

45 

0   20  52 

0   21   15 

o  21  40 

O   22  06 

22  33 

0   23  02 

—  19 

oo 

o  27  40 

0   28   II 

o  28  44 

o  29  18 

29  55 

3o  33 

—  25 

*5 

o  34  21 

o  34  59 

o  35  40 

o  36  23 

37  08 

37  56 

—  32 

30 

o  40  52 

o  41  38 

o  42  26 

o  43  17 

44  ii 

45  08 

•  -38 

45 

o  47  12 

o  48  05 

o  49  01 

0  49  59 

51  02 

52  07 

—  43 

oo 

o  53  *9 

0  54  '9 

o  55  22 

o  56  28 

57  38 

58  52 

—  49 

15 

o  59  12 

I   00   l8 

I   OI   28 

i  02  41 

°3  59 

05   21 

—  54 

30 

i  04  49 

i  06  01 

I  07  17 

i  08  38 

10  03 

II   32 

—  59 

2  45 

I   10  08 

I  II  26 

I   12  48 

1   !4   15 

15  47 

17   24 

-64 

3  oo 

i  15  09 

i  16  32 

i  18  oo 

1  J9  33 

21   II 

22  54 

—  68 

3  15 

i  19  49 

I   21   17 

I   22   50 

i  24  29 

26   I3 

28   02 

—  72 

3  3<> 

i  24  08 

I  25  40 

i  27  18 

i  29  02 

30  51 

32   46 

—  76 

3  45 

i  28  04 

I  29  41 

1  31  23 

i  33  ii 

35  °5 

37  °6 

—  80 

4  oo 

i  3i  37 

i  33  17 

i  35  03 

i  36  55 

38  54 

40  59 

—  83 

4  IS 

i  34  45 

i  36  29 

i  38  18 

i  40  14 

42  16 

44  *5 

—  86 

4  30 

i  37  29 

i  39  15 

i  41  08 

i  43  06 

45  " 

47  24 

—  88 

4  45 

i  39  47 

i  4i  35 

i  43  30 

1  45  31 

47  39 

49  54 

—  90 

5  oo 

t  41  38 

i  43  29 

1  45  25 

i  47  28 

49  38 

Si  55 

—  91 

5  15 

i  43  °4 

1  44  55 

i  46  53 

1  48  57 

51  08 

53  27 

—  92 

5  30 

i  44  02 

i  45  54 

1  47  53 

i  49  58 

52  10 

54  3° 

—  93 

5  45 

i  44  34 

i  46  26 

i  48  25 

i  50  30 

52  43 

55  03 

—  94 

6  oo 

i  44  38 

i  46  31 

i  48  29 

1  So  34 

5*  46 

55  06 

—  93 

6  15 

i  44  16 

i  46  08 

i  48  05 

i  50  10 

52   21 

54  40 

—  93 

6  to 

i  43  27 

i  45  18 

i  47  14 

i  49  17 

I   5I   27 

i  53  44 

—  92 

6  45 

I   42   12 

i  44  01 

i  45  56 

1  47  56  • 

I   5"  °4 

I   52   20 

—  91 

7  oo 

i  40  31 

i  42  18 

i  44  10 

i  46  09 

I   48   M 

I   50  27 

-89 

7  15 

i  38  24 

i  40  09 

i  4i  59 

1  43  54 

i  45  57 

I   48  06 

—  87 

7  3° 

i  35  53 

i  37  35 

i  39  21 

i  41  14 

i  43  13 

i  45  19 

-85 

7  45 

:  32  58 

i  34  36 

i  36  19 

i  38  08 

i  40  03 

i  42  05 

—  82 

8  oo 

i  29  40 

i  3i  M 

i  32  53 

1  34  38 

i  36  29 

i  38  26 

—  79 

8  15 

i  25  59 

i  27  29 

i  29  04 

1  30  44 

i  32  30 

i  34  22 

-76 

8  30 

i  21  57 

i  23  23 

i  24  53 

i  26  28 

i  28  09 

i  yg  55 

—  72 

8  45 

i  17  35 

i  18  56 

I   20   21 

1   21   51 

i  23  26 

i  25  07 

—  68 

9  oo 

i  12  54 

i  14  10 

I  15  30 

i  16  54 

i  18  23 

i  19  57 

—  64 

9  15 

i  07  54 

i  09  05 

i  10  19 

i  ii  38 

i  13  01 

i  14  28 

—  59 

9  3° 

I   02  38 

i  03  44 

I  04  52 

i  06  04 

I   07  21 

i  08  41 

—  55 

9  45 

o  57  07 

o  58  07 

o  59  09 

i  oo  15 

I   OI   24 

I   02   38 

—  50 

TO   00 

0   51   22 

o  52  16 

o  53  I2 

0  54  " 

o  55  X3 

o  56  19 

—  45 

10  15 

o  45  25 

0   46   12 

o  47  01 

o  47  53 

o  48  49 

o  49  47 

—  40 

10  30 

10  45 

II   00 

o  39  16 
o  32  58 
o  26  32 

0  39  57 
o  33  32 
o  27  oo 

o  40  40 
o  34  08 
o  27  28 

o  41  25 

o  34  46 
o  27  59 

o  42  12 
o  35-  26 
o  28  31 

o  43  02 
o  36  08 
o  29  05 

—  34 
—  29 
—  2| 

II  15 

o  20  oo 

o  20  20 

0   20   42 

0   21  05 

O   21   29 

o  21  55 

—  18 

II  30 

11  45 

o  13  23 
o  06  42 

o  13  36 
o  06  49 

o  13  51 

o  06  56 

o  14  06 
o  07  04 

0   14  22 
0   07  12 

o  14  39 

0   07   21 

—  6 

Elongation: 

Azimuth  

i  44  40 

i  46  32 

I  48  31 

i  50  36 

I   S2  48 

i  55  08 

—  93 

h.  m.  S. 

h.  m.  s. 

h.  nt.  s. 

i.  nt.  s. 

h.  m.  s. 

h.  nt.  s. 

s. 

Hour-angle. 

5  55  »4 

5  54  53 

5  54  42 

5  54  3i 

5  54  20 

5  54  07 

+  5 

326 


GEODETIC  ASTRONOMY. 


§310. 


Correctionfor 

Hour-angle 
before  or  after 
Upper 

AZIMUTH  OF  POLARIS  COMPUTED  FOR 
DECLINATION  88°  46'. 

i'  Increase  in 
Declination 
of  Polaris. 

Culmination. 

Lat.  50°. 

Lat.  51°. 

Lat.  52°. 

Lat.  53°. 

Lat.  54°. 

Lat.  55°. 

Lat.  50°. 

"T 

o°  07'  44" 

o'o7'54< 

o°  08'  05' 
o  16  08 

o°  08'  17' 

o°  08'  29" 

o°  08'  42' 

—  6" 

45 

o  23  02 

o  23  33 

o  24  06 

o  24  41 

o  25  18 

0  25  57 

—  19 

00 

0  3°  33 

o  31  14 

o  31  58 

o  32  44 

o  33  33 

o  34  25 

—  25 

15 

o  37  56 

o  38  47 

o  39  40 

o  40  38 

o  41  38 

o  42  43 

—  32 

3° 

o  45  08 

46  08 

o  47  12 

0   48  20 

o  49  32 

o  50  49 

-38 

45 

o  5*  07 

53  J7 

o  54  3i 

o  55  49 

o  57  12 

o  58  41 

—  43 

oo 

o  58  52 

OO  II 

i  01  34 

i  03  03 

1  04  37 

i  06  16 

—  49 

15 

I   05  21 

06  48 

i  08  21 

i  09  59 

i  ii  43 

1  J3  33 

—  54 

3° 

I   II   32 

13  08 

i  14  48 

i  16  35 

i  18  29 

i  20  30 

—  59 

2  45 

I   17  24 

19  07 

i  20  55 

i  22  51 

i  24  54 

i  27  04 

-64 

3  oo 

i  22  54 

24  44 

i  26  41 

i  28  44 

i  3°  55 

*  33  15 

—  68 

3  15 
3  3° 

I   28  02 

I   32  46 

29  59 
34  49 

I   32   02 

I   36   58 

i  34  i| 
i  39  16 

i  36  32 
i  41  42 

i  39  oo 
i  44  18 

—  72 

-76 

3  45 

i  37  06 

39  »4 

I   4I   29 

43  52 

i  46  25 

i  49  07 

—  80 

4  oo 

i  40  59 

43  12 

i  45  32 

48  01 

i  So  39 

53  27 

-83 

4  15 

i  44  25 

46  42 

i  49  07 

51  40 

1  54  23 

57  16 

-86 

4  3° 

i  47  24 

49  44 

i  5»  *3 

54  50 

i  57  37 

oo  35 

—  88 

4  45 

1  49  54 

52  17 

i  54  49 

57  29 

2   OO  2O 

03  21 

—  90 

5  oo 

*  5^  55 

54  21 

i  56  54 

59  37 

2   02   31 

°5  35 

—  91 

5  i5 

*  53  27 

55  54 

i  58  29 

01  15 

2   04   10 

07  16 

—  92 

5  3° 

1  54  3° 

56  58 

i  59  34 

02   20 

2  05  16 

08  23 

—  93 

5  45 
6  oo 
6  15 

*  55  03 
i  55  06 
i  54  4° 

57  3* 
57  34 
i  57  06 

2   00  08 
2   00  10 

i  59  4i 

02  53 

02   56 

02  26 

2   05  50 
2   05  52 
2   05  21 

08  58 
08  58 
08  26 

—  94 
—  93 
—  93 

6  45 
7  oo 

1  53  44 

I   52  20 

I   50  27 

i  54  42 
i  52  47 

i  57  14 
55  IS 

i  59  54 
i  57  52 

2   02  44 

2  oo  39 

2  05  45 

2   03   36 

—  92 
—  91 
—  89 

7  15 

1   48  06 

i  5°  23 

52  48 

i  55  21 

I   58  04 

2  oo  57 

-87 

7  30 

i  45  19 

i  47  32 

49  52 

i  52  21 

i  54  59 

1  57  47 

-85 

7  45 

i  42  05 

i  44  13 

46  29 

i  48  53 

i  51  26 

i  54  08 

—  82 

8  oo 

i  38  26 

i  40  29 

42  40 

i  44  58 

1  47  25 

i  50  01 

—  79 

8  15 

i  34  22 

I   36   20 

38  25 

i  40  38 

i  42  58 

I  45  27 

—  76 

8  30 

1  29  55 

I   3I   48 

33  47 

i  35  52 

i  38  06 

i  40  28 

—  72 

8  45 

i  25  07 

i  26  53 

28  45 

1  30  44 

1  32  5° 

i  35  04 

—  68 

9  oo 

»  '9  57 

i  21  37 

23  22 

i  25  13 

i  27  ii 

i  29  17 

-64 

9  *5 

i  14  28 

i  16  01 

17  38 

I   19   22 

I   21   12 

i  23  08 

—  59 

9  30 
9  45 

i  08  41 

I   02   38 

10  06 

03  55 

ii  36 
05  17 

I   I|   12 

i  06  44 

1  *4  53 
i  08  16 

i  16  40 
i  09  53 

—  55 
—  So 

10   00 

o  56  19 

o  57  28 

o  58  42 

I   00  00 

i  01  23 

I   02  50 

—  45 

10  15 

o  49  47 

o  50  48 

o  5i  53 

o  53  02 

°  54  IS 

o  S5  32 

—  40 

10  30 
10  45 

o  43  02 
o  36  08 

3  43  56 
o  .36  52 

o  44  S2 
o  37  39 

o  45  51 
o  38  29 

°  46  54 
o  39  22 

o  48  01 
o  40  18 

—  34 
—  29 

It   00 

o  29  05 

o  29  41 

o  30  18 

o  30  58 

o  31  41 

o  32  26 

—  23 

ii  15 

o  21  55 

0   22  22 

O   22   50 

0   23  20 

o  23  52 

o  24  26 

—  18 

it  30 

o  14  39 

0   14  57 

o  15  16 

o  15  37 

o  15  58 

o  16  21 

—  12 

Elongation  : 
Azimuth  

o  07  21 
i  55  08 

o  07  30 
i  57  36 

o  07  39 

2   OO  13 

o  07  49 
2  02  59 

o  08  oo 
2  05  55 

o  08  ii 

',  09  02 

-  6 
—  93 

ft  tn   s 

I  Wl    S 

h.  tn   s 

t.  Wl   S. 

t  tn   s 

t  tn  s 

Hour-angle. 

5  54  «>7 

5  53  54 

5  53  4» 

5  53  27 

5  53  « 

S  52  57 

+  5 

TABLES. 


327 


Correction  for 

Hour-angle 
before  or  after 

AZIMUTH  OF  POLARIS  COMPUTED  FOR 
DECLINATION  88°  46'. 

i'  Increase  in 
Declination 
of  Polaris. 

Upper 

Culmination. 

Lat.  55°. 

Lat.  56°. 

Lat.  57°. 

Lat.  58°. 

Lat.  59°. 

Lat.  60°. 

Lat.  60°. 

o"  is" 

o°  08'  42" 

o°  08'  56" 

o*  09'  12" 

o°  09'  28" 

o°  09'  45" 

0  io'  03" 

—  8" 

o  30 

o  17  22 

o  17  50 

0   l3  20 

o  18  53 

o  19  27 

20  04 
on  cH 

—  17 

o  45 

I   00 

o  25  57 
o  34  25 

o  20  39 
o  35  21 

0   36   20 

o  37  23 

o  38  31 

29  5° 
39  44 

25 
—  33 

I  15 

o  42  43 

o  43  52 

o  45  06 

o  46  24 

o  47  48 

49  19 

I  30 

o  50  49 

o  52  ii 

o  53  39 

o  55  12 

o  56  52 

58  40 

—  49 

1  45 

2   00 

o  58  41 
i  06  16 

i  oo  16 
i  08  03 

i  oi  50 
i  09  57 

i  03  44 
i  ii  58 

i  05  40 
i  14  08 

07  44 
16  28 

-35 

2   15 

i  13  33 

1  *5  31 

i  17  37 

i  19  52 

I   22  l6 

i  24  51 

—  71 

2   30 

I   20  30 

i  22  39 

i  24  56 

i  27  24 

i  30  oi 

32  So 

-  78 

2  45 

I   27  04 

i  29  23 

i  31  52 

i  34  3i 

i  37  21 

40  23 

T     At    oft 

80 

3  °° 
3  J5 

1  33  X5 
i  39  oo 

1  35  43 
i  4i  37 

i  44  25 

i  47  25 

i  5°  37 

47  2O 
i  54  °3 

-••  09 

—  94 

3  30 

i  44  18 

1  47  03 

i  50  oo 

i  53  08 

i  56  30 

2   00  07 

—  99 

3  45 
4  oo 

1  53  27 

i  56  26 

1  55  °4 
1  59  37 

2   03  OI 

2   06  40 

2  05  37 
2  io  34 

—  108 

4  15 

i  57  16 

2   00  21 

2  03  38 

2   07  09 

2  io  54 

2  14  55 

—  in 

4  30 
4  45 

2  oo  35 

2   03  21 

2  03  44 

2  06  34 

2   07  00 
2   IO  OO 

2   IO  42 
2   13  40 

2   I4  32 

2  17  35 

2  18  39 
2  21  47 

-114 
—  116 

5  oo 

2  05  35 

2   08  51 

2   12  20 

2  16  03 

2   2O  O2 

2  24  17 

—  118 

5  IS 

2  07  16 

2  io  34 

2   14  05 

2   17   50 

2   21   51 

2   26  09 

—  119 

5  30 

2   08   23 
2   08   58 

2   II   42 

2   15   I4 

2   19  01 

2   23  04 

2   2  1   QQ 

2   27  2| 

2   27   t;8 

120 
I2O 

6  oo 

2   08  58 

2   12   17 

2  15  49 

2  19  35 

•*  •*  o  jy 
2  23  37 

•*  •*/  5° 

2   27  56 

120 

6  15 

2   08   26 

2  ii  44 

2   15   14 

2  18  59 

2   22   59 

2   27  15 

—  II9 

6  30 

2   07   22 

2  io  37 

2   14  05 

2  17  47 

2  21  44 

2  25  57 

—  118 

6  45 

2  05  45 

2  08  57 

2   12   21 

2   l6  00 

2  19  53 

2   24  03 

—  116 

7  oo 

2   03   36 

2  06  44 

2   10  05 

2  13  39 

2   17  27 

2   21   32 

—  114 

7  IS 

2  oo  57 

2   04  00 

2  07  16 

2  io  45 

2   14  27 

2  18  26 

—  in 

7  3° 

i  57  47 

2  oo  45 

2  03  55 

2  07  18 

2  io  54 

2   14   46 

—  108 

7  45 

i  54  08 

i  57  oo 

2   00  04 

2   03  20 

2  06  49 

2   10  32 

—  104 

8  oo 

i  50  oi 

i  52  47 

1  55  43 

I   58  52 

2   02  12 

2  05  47 

IOO 

8  15 

i  45  27 

i  48  06 

i  So  54 

i  53  54 

i  57  06 

2   00  32 

-  96 

8  30 

i  40  28 

i  42  58 

1  45  39 

i  48  30 

i  51  32 

i  54  47 

—  91 

8  45 

1  35  04 

i  37  26 

1  39  57 

i  42  39 

i  45  3i 

i  48  35 

—  86 

9  oo 

i  29  17 

i  31  30 

i  33  5i 

i  36  23 

i  39  05 

1  4i  57 

—  80 

9  15 
9  30 

i  23  08 
i  16  40 

I   25   12 

i  18  34 

i  27  24 

I   20   36 

i  29  44 
i  22  45 

i  32  14 
i  25  03 

*  34  55 
i  27  30 

—  75 
-  69 

9  45 

*  09  53 

1  IT  37 

I   13   28 

i  15  25 

i  17  31 

i  19  45 

63 

10   00 

I   02   50 

i  04  23 

I   06  03 

i  07  48 

i  09  41 

i  ii  41 

—  56 

io  15 

o  55  32 

o  56  54 

O   58  22 

o  59  55 

i  oi  34 

i  03  20 

—  50 

io  30 

o  48  oi 

o  49  12 

o  50  27 

o  51  48 

o  53  14 

o  54  45 

—  43 

io  45 

o  40  18 

o  41  18 

0   42  21 

o  43  28 

o  44  40 

o  45  57 

-  36 

II   00 

o  32  26 

o  33  M 

o  34  05 

o  34  59 

o  35  57 

o  36  59 

—  29 

II  15 

o  24  26 

0   25  02 

o  25  41 

0   20  21 

o  27  05 

o  27  51 

22 

ii  3° 

o  16  21 

o  16  45 

o  17  io 

o  17  38 

o  18  07 

o  18  38 

—  14 

«  45 
Elongation  : 

o  08  ii 

o  08  23 

o  08  36 

o  08  50 

o  09  04 

o  09  20 

—   7 

Azimuth... 

2   09  O2 

2   12  21 

2  15  54 

2   19   40 

2  23  43 

2   28  O2 

I2O 

Jt   9tt 

L   jfl    f 

k.  nt.  s 

h.  m.  s 

h  nt  s 

h  in  s 

- 

Hour-angle 

5  52  57 

5  52  41 

5  52  24 

5  52  06 

5  51  47 

5  51  27 

-f   7 

328  GEODETIC  ASTRONOMY.  §3'I- 


311.  NOTATION   AND  PRINCIPAL  WORKING 
FORMULAE. 

The  following  general  notation  is  used  throughout  the 
book.  The  special  notation  involved  in  each  working  for- 
mula will  be  found  below  each  group  of  formulae. 

GENERAL    NOTATION. 

a  and  3  =  the  apparent  right  ascension  and  declination, 
respectively,  at  the  time  of  the  observation 
under  consideration. 

am  and  dm  =  the   mean   right   ascension   and  declination,   re- 
spectively. 

al87S  and  tfl875  (with  a  year  as  a  subscript)  =  the    mean    right 
ascension  and  declination,  respectively,  at  the 
beginning  of  the  fictitious  year  indicated. 
<*0  and  #0  =  the  values  of  am  and  $m  at  the  beginning  of  the 
fictitious  year  during  which  the  observation 
under  consideration  was  made. 
}*  and  X  =  proper  motions,  per  year,  in  right  ascension  and 

declination,  respectively. 
A  =  altitude. 
C  =  zenith  distance. 

0  =  astronomical  latitude  of  the  station  of  observa- 
tion. 

t  =  hour-angle,  measured  eastward  or  westward  from 
the  upper  branch  of  the  meridian  as  the  case 
may  be,  but  always  considered  positive,  and 
never  exceeding  180°  (or  I2h). 


§3IX-  NOTATION  AND    WORKING   FORMULAE.  329 

z  =  the  azimuth  of  a  star,  measured  to  the  eastward 
or  westward  from  north  as  the  case  may  be, 
but  always  considered  positive  and  never 
exceeding  180°. 

d  =  value,  in  arc,  of  one  division  of  a  level. 


WORKING    FORMULA, 

WITH  THEIR  SPECIAL  NOTATION,  AND  WITH  REFERENCES 
TO  CORRESPONDING  PORTIONS  OF  THE  TEXT  AND  TO 
THE  TABLES. 

To  convert  mean  solar  to  sidereal  time. 

See  §  23  and  the  table  of  §  290. 
To  convert  sidereal  to  mean  time. 

See  §  24  and  the  tables  of  §§  290,  291. 
To  interpolate  along  a  chord. 

,Vi-V. 


or 

Fj  =  Ft  —  (F9  —  Ft)    *  _    y  ;  .     .     .     .     (i)  * 

the  first  form  being  used  when  the  interpolation  is  made 
forward  from  the  value  Flt  and  the  second  when  it  is  made 
backward  from  the  value  F9.  F7  is  the  required  interpolated 
value  of  the  function  corresponding  to  the  value  Vt  of  the 
independent  variable.  V^  and  Fa  are  the  adjacent  stated 
values  of  the  independent  variable  to  which  correspond  the 
given  values  F^  and  Ft  of  the  function.  See  §  30. 

*  The  number  assigned  to  each  formula  corresponds  to  that  used  in  the 
body  of  the  text. 


33O  GEODETIC  ASTRONOMY.  §  311. 

To  interpolate  along  a  tangent. 


FI  is  the  required  interpolated  value  of  the  function  corre- 
sponding to  the  value  F/  of  the  independent  variable.  F1  and 
Fl  are,  respectively,  the  nearest  given  value  of  the  independ- 
ent variable,  and  the  corresponding  value  of  the  function. 

f—  —  J    is  the  given  first  differential  coefficient  corresponding 

to  V,.     See  §  31. 

To  interpolate  along  a  parabola. 

If  the  first  differential  coefficients  are  given, 

KdF\ 
^)1 

or 


according  to  whether  the  interpolation  is  made  forward  from 
Vl  or  backward  from  F2.  Fj  is  the  required  interpolated  value 
of  the  function  corresponding  to  the  value  F/  of  the  inde- 
pendent variable.  F,  and  F,  are  the  adjacent  stated  values  of 
the  independent  variable  to  which  correspond  the  given 

(dF\ 
values  Fl  and  /%  of  the  function,  and  the  given  values  \-Jyf 

and  \-jp>)    of  the  first  differential  coefficient.     See  §  33. 
If  the  first  differential  coefficients  are  not  given, 


gS11-  NOTATION  AND    WORKING  FORMULAE.  331 

or 


~  V,  +  D)\  [  V,  -  F,]. 


(4)  is  the  general  formula  which  is  applicable  even  when 
the  successive  differences  between  Vlf  Fa,  Fs,  .  .  .  are  not  all 
the  same,  and  (40)  is  the  formula  for  the  special  case  in 
which  those  differences  are  all  the  same. 

F[  is  the  required  interpolated  value  of  the  function  cor- 
responding to  the  value  F/  of  the  independent  variable. 
Flt  FV  and  Fs  are  three  successive  given  values  of  the  function 
corresponding,  respectively,  to  the  values  Fp  F,,  F,  of  the 
independent  variable,  Fa  being  the  stated  value  of  the  variable 
nearest  to  which  lies  the  value  F/. 

In  (4*)  D  =  (FQ  -  FO  =  (F3  -  Fa)  =  .  .  .  ,  and  </,  is  the 
second  difference,  or  (F,  -  F,)  -  (Ft  -  F,).  See  §  34. 

Having  given  the  mean  place,  am  and  dm,  of  a  star  for  a 
date  tm  at  the  beginning  of  some  fictitious  year,  to  compute  its 
mean  place,  #0,  £0,  at  /o,  the  beginning  of  the  fictitious  year 
during  which  the  observations  under  consideration  were  made. 


(8) 
;  •    (9) 


in  which  t0  and  tm  are  expressed  in  years,  -^  is  the  change 

7  ^  //*/t 

in  «w  per  year,  -^  is  the  change  in  dm  per  year,  and  —^  is 

7^ 

the  change  in  —     per  year.      Lists  of  star  places  usually  give 


332  GEODETIC  ASTRONOMY.  §  311. 

the  numerical  values  of  —  ~  and  —  rr,  and  sometimes      .  ./",  for 

at  at  at 

y  a  «  r  i<* 

each  star.      —  ^  is  tabulated  in  §  292.     (N.B.  -~  and  —£• 

include  both  the  effect  of  precession  and  of  proper  motion.) 
See  §§  40-43- 

Having  given  a0  and  tf0,  the  mean  place  of  a  star  at  the 
beginning  of  the  fictitious  year  during  which  the  observations 
under  consideration  were  made,  to  compute  the  apparent  place, 
a  and  dy  at  the  instant  of  observation. 


sin  (G  +  or0)  tan  tfo 
sin  (7/+  "o)  sec  d0   ......   (in  time);    (10) 


+  //  cos  (//  +  <x0)  sin  tf0  +  i  cos  tf0  .   .   (in  arc);      (i  i) 

in  which/,  G,  H,  g,  h,  and  /  are  quantities  called  independ- 
ent star-numbers  which  are  functions  of  the  time  only  and  are 
given  in  the  Ephemeris  for  every  Washington  mean  midnight. 
Their  values  for  the  instant  of  observation  may  be  derived'  by 
interpolations  along  chords  between  the  Ephemeris  values. 
r  is  the  elapsed  portion  of  the  fictitious  year  expressed  in 
units  of  one  year.  It  is  given  in  the  Ephemeris  with  the  star- 
numbers.  See  §§  46-49. 

To  compute  the  correction  to  a  timepiece  on  mean  time  from 
observations  of  the  double  altitude  of  the  Sun  with  a  sextant 
and  artificial  horizon.  See  §§  62-70. 

The  mean  reading  of  the  sextant  arc  corrected  for  index 
error  and  eccentricity  is  2AU.  Au  is  the  approximate  altitude 
of  the  Sun.  For  the  method  of  correcting  for  index  error  see 
§  62,  and  for  eccentricity  see  §  76.  The  altitude 

A  =  Au  ±  Sun's  semi-diameter  -\-p-R. 


§3H.  NOTATION  AND    WORKING   FORMULAE.  333 

The  Sun's  semi-diameter,  as  taken  from  the  Ephemeris,  is 
to  be  added  if  the  Sun's  lower  limb  was  observed,  and  sub- 
tracted if  the  upper  limb  was  observed. 

The  parallax,  /,  is  given  in  the  table  of  §  293. 

The  refraction  is  R  =  RM(CB)(CD)(CA).  RM,  CB,  CD,  and 
CA  are  given  in  the  tables  of  §§  294-297.  The  refraction  is 
required  for  the  altitude  Aul  not  A. 

C  -  90°  -  A. 

sin  j[C  +  (0  -  *)]  sin  j[C  -  (0  -  tf)] 


_ 
1  i/=  cos      cos  tf 


See  general  notation.  d  may  be  obtained  from  the 
Ephemeris  by  interpolation  along  a  tangent. 

I2h  i  /  =  apparent  solar  time  =  7^. 

TA  -\-  £  =  mean  solar  time  =  TM. 

E  is  the  equation  of  time  which  is  given  in  the  Ephemeris 
for  Washington  apparent  noon,  and  may  be  obtained  for  the 
instant  of  observation  by  interpolation  along  a  chord. 

Te  =  mean  reading  of  timepiece. 

TM  —  Tc  =  A  Tc  =  required  correction  to  timepiece. 

If  the  observations  are  taken  at  sea  using  the  natural  hori- 
zon,  the  mean  reading  of  the  sextant  arc  corrected  for  index 
error  and  eccentricity  is  Au  (not  2AU). 

A  =  Au  ±  Sun's  semi-diameter  +/  —  R  —  Dip. 

The  first  four  terms  in  the  second  member  are  the  same 
as  before.  The  dip,  or  downward  inclination  of  the  line  of 
sight  to  the  apparent  horizon  due  to  the  height  of  the  sextant 
above  the  surface  of  the  sea,  is  given  in  the  table  in  §  298. 
The  remainder  of  the  computation  is  as  given  above. 


334  GEODETIC  ASTRONOMY.  §3H» 

TO  COMPUTE  THE  CORRECTION  TO  A  SIDEREAL  TIMEPIECE 
FROM  OBSERVATIONS  WITH  AN  ASTRONOMICAL  TRANSIT 
PLACED  IN  THE  MERIDIAN.  §§  90-99. 

If  the  times  of  transit  of  a  star  across  some  (but  not  all) 
of  the  lines  of  the  reticle  were  observed,  the  time,  tmy  of 
transit  across  the  mean  line  of  the  reticle  may  be  computed 
by  the  formula 

(sum  of  equatorial  intervals 
of  observed  lines)  (sec  5) 

tm  =  mean  of  observed  times : — : — — r= *    •     (25) 

number  of  observed  lines 


or 


(sum  of  equatorial  intervals 
of  missed  lines)  (sec  d) 

tm  =  mean  of  observed  times  H -e — r~r> •    •     (26) 

number  of  observed  lines 


For  the   process  of  finding  the  equatorial   intervals  see 


*8 


in  which  Te'  is  the  observed  time  of  transit  across  the  mean 
line  corrected  for  inclination  of  the  horizontal  axis  and  for 
diurnal  aberration. 

B  —  cos  C  sec  d  is  tabulated  in  §  299. 

b  =  fi  ±  /,-,  /,•  being  the  pivot  inequality  derived  as 
indicated  in  §§94,  115. 

fi=\(w+uS)-(e  +  S)\£      .     .     .     (29) 

if  the  level  divisions  are  numbered  from  the  middle  toward 
each  end,  w  and  w'  being  the  west  end  readings  of  the  bubble 


§3IX'  NOTATION  AND    WORKING   FORMULA.  335 

before  and  after  reversal  of  the  striding-level,  and  e  and  e'  the 
corresponding  east  end  readings. 


/*={(«,+  ,)_  (w1  +  e')\  .     .      .     (30) 

if  the  level  divisions  are  numbered  continuously  from  one  end 
to  the  other,  the  primed  letters  indicating  the  readings  taken 
with  the  zero  of  the  level  to  the  westward. 
k  is  tabulated  in  §  315. 

Tc  =  77+  Aa  +  Cc, 

in  which  Tc  is  the  reading  of  the  timepiece  when  a  star  crosses 
the  meridian,  and  Aa  and  Cc  are  the  corrections  for  azimuth 
error  and  collimation  error,  respectively. 

A  =  sin  £  sec  #  and  C  =  sec  d  are  tabulated  in  §  299. 

To  compute  the  azimuth  and  collimation  errors  (a  and  c) 
from  the  observations,  WITHOUT  THE  USE  OF  LEAST  SQUARES 
(§  101-106),  use  the  formulae 

(a-T;\v-(a-  TC'}E 


\Ot          lc  6 £)time  stars          \a          *c  ^-^)  azimuth  star      /       x 

a~ ~^~~  ~A~  >  (40 

•'-1  time  stars          •ri  azimuth  star 

to  derive  a  and  c,  by  successive  approximations,  as  indicated 
in  §§  101-106. 

The  clock  correction  as  determined  by  each  observation  is 
ATC=  a—  T. 

To  compute  the  azimuth  and  collimation  errors  (a  and  c), 
and  ATC,  from  the  observations,  BY  LEAST  SQUARES.  (§§ 
107-110.) 


GEODETIC  ASTRONOMY.  §  311. 

The  observation  equations  are  of  the  form 

Awaw+Cc-(a-  27)  =  o,     .     .     (45) 


and 

ATC  +  AEaE  +  Cc  -  (a  -  Tc')    =  o,     .     .     (46) 

for  the  observations  made  with  illumination  west  and  east, 
respectively. 

The  normal  equations  are 


SA  waw  +  2AsaB    +  2Gc       -  2(a  -  Te')      =  o; 
+  2AwCc-2AuAa-Tc')=o; 
^A*EaE    +  2ABCc  -  2A£(a  -  Tc')  =  o', 


-2C(a-Tc')    =  o. 

The  solution  of  these  four  equations  gives  the  values  of 

j  and  c. 
The  probable  error  of  a  single  observation  is 


/    2v* 

=  0.674*  / , 

V  ».  -  nn 


in  which  the  v's  are  the  residuals  of  the  observation  equations, 
n0  is  the  number  of  observations,  and  nu  is  the  number  of 
unknowns  (and  of  normal  equations). 

The  probable  error  of  the  computed  ATC  is  e0  =  e  V Q,  in 
which  Q  is  a  quantity  obtained  as  follows:  In  equation  (47) 
write  Q  in  the  place  of  ATC,  —  i  in  the  place  of  2(a  —  T/), 
and  o  in  the  place  of  the  other  absolute  terms,  and  then  solve 
for  Q. 

For  two  modifications  of  this  method  of  computing,  which 
may  be  used  if  considered  advisable,  see  §§  109,  no. 

For  the  form  of  computation  if  unequal  weights  (depend- 


§3TI-  NOTATION  AND    WORKING  FORMULA.  337 

ing  upon  the  declination  of  the  star  and  the  number  of  lines 
of  the  reticle  observed  upon)  are  assigned  to  the  separate 
observations,  see  §§  111-113. 

TO  COMPUTE  THE  LATITUDE   FROM   OBSERVATIONS  MADE 
WITH   A   ZENITH    TELESCOPE.      (§§   146-157.) 

The  latitude  from  a  single  pair  of  stars  is 


-  M')~  +  d-\(n  +  n')  -  (s 


-  '..    (57) 


In  (57)  the  primed  letters  correspond  to  the  northern  star 
of  the  pair;  M  is  the  micrometer  reading  expressed  in  turns;  r 
is  the  angular  value  of  one  turn ;  n  and  s  are  the  north-end  and 
south-end  readings,  respectively,  of  the  level,  for  the  northern 
star,  and  n'  and  s1  for  the  southern  star;  R  is  the  refraction, 
and  m  the  reduction  to  the  meridian  of  a  star  observed  off  the 
meridian. 

The  level  correction  as  given  above  is  for  a  level  tube 
which  carries  a  graduation  of  which  the  numbering  increases 
each  way  from  the  middle.  If  the  level-tube  graduation  is 
numbered  continuously  from  one  end  to  the  other  the  level 

correction  becomes  ~S(X  +  s')  ~  (n  +  s)\-     ^See  §  X48.) 
The  term  %(R  —  R')  is  tabulated  in  §  304.     (See  §  150.) 
The  term  —  is  tabulated  in  §  305.     (See  §  151.) 


338  GEODETIC  ASTRONOMY.  §31 1- 

To  combine  the  separate  values  of  0  and  to  compute  the 
probable  errors.      (§§  154-157.) 

The  probable  error  of  a  single  observation  is 


/v^s-a^H,.  .....    (64) 


in  which  the  J's  are  the  differences  obtained  by  subtracting 
the  mean  result  for  each  pair  from  the  result  on  each  separate 
night  from  that  pair;  [JJ]  is  the  sum  of  the  squares  of  the 
J's;  n  is  the  total  number  of  observations;  and/  is  the  total 
number  of  pairs  observed. 

The  probable  error  of  the  mean  result  from  any  one  pair 
is 


(o.45  5  ) 


in  which  the  v's  are  the  residuals  obtained  by  subtracting  the 
indiscriminate  mean  result  for  the  station  from  the  mean 
result  from  each  pair;  and  [vv\  is  the  sum  of  the  squares  of 
the  7/s. 

The  probable  error  of  the  mean  of  the  two  declinations  of 
the  stars  of  a  pair  is 


t,  =  «/  -  e',    ......     (68) 

in  which 


§3XI-  NOTATION  AND    WORKING   FORMULA.  339 

nlt  n^  #,,...  are  the  numbers  of  times  that  pair  No.  I,  pair 
No.  2,  pair  No.  3,  .  .  .  ,  respectively,  are  observed. 

The  proper  weights  wlt  wt,  wif  .  .  .  for  the  mean  results 
from  the  separate  pairs  are  proportional  to 


ns 

The    most    probable  value,    00,    for  the  latitude  of  the 
station  is 

'3  .-.       O0]  ,„. 

=  7^T»    •      •     (7°) 


in  which  <pl  is  the  mean  result  from  the  first  pair,  0,  from  the 
second  pair,  and  so  on. 

The  probable  error  of  00  is 


(0. 

^' 


in  which  [wz;/2]  stands  for  the  sum  of  the  products  of  the 
weight  for  each  pair  into  the  square  of  the  residual  obtained 
by  subtracting  00  from  the  mean  result  for  that  pair;  and  [w] 
is  the  sum  of  the  weights. 

Computation  of  the  Micrometer  Value  from  Observations 
upon  a  Circumpolar  Star  near  Elongation.  (§§  158-162.) 

For  an  example  of  this  computation  see  §  162. 

To  compute  the  hour-angle  of  the  star  at  elongation  use 
the  formula 

cos  tB  =  tan  0  cot  <?,   .         ...     (72) 


34O  GEODE  TIC  A  S  TRONOM  Y.  §  3 1 1 . 

ts  added  to  or  subtracted  from  the  right  ascension  of  the 
star,  for  a  western  or  eastern  elongation  respectively,  gives 
the  sidereal  time  of  elongation. 

The  correction  for  curvature  is  given  in  §  306. 

The  level  correction  is  given  by  formulae  (77)  and  (78)  of 

§  1 60. 

If  /  is  the  time  interval  in  seconds  corresponding  to  one 
turn  of  the  micrometer,  then  the  value  of  one  turn  expressed 
in  seconds  of  arc  is  (15  cos  $)t.  (See  §  161.) 

To  this  value  must  still  be  applied  the  corrections  for 
chronometer  rate  and  for  refraction  as  indicated  at  the  end  of 
§  161. 

To  Compute  the  Latitude  from  an  Observed  Altitude  of  a 
Star,  or  the  Sun,  in  any  position,  the  time  being  known. 
See§  171. 

To  Compute  the  Latitude  from  Zenith  Distances  of  a  Star, 
or  the  Sun,  Observed  near  the  Meridian,  the  time  being  known. 
See  §  172. 

To  Compute  the  Latitude  from  Observations  of  the  Altitude 
of  Polaris  at  any  Hour-angle,  the  time  being  known. 
See  §  173. 

To  compute  azimuth  from  observations  upon  a  circumpolar 
star  with  a  direction  instrument. 

Example  of  record,  §  187. 
Example  of  computation,  §  200. 
Level  correction 


,      .     .     (91) 


§3IX'  NOTATION  AND    WORKING   FORMULA.  34! 

for  a  level  having  its  divisions  numbered  both  ways  from  the 
middle. 


(92) 


for  a  level  numbered  continuously  in  one  direction,  the 
primed  letters  referring  to  the  readings  taken  in  the  position 
in  which  the  numbering  increases  toward  the  east.  CL  as 
given  by  these  formulae  is  the  correction  to  the  circle  reading 
for  the  star  upon  the  supposition  that  the  circle  graduation 
increases  in  a  clockwise  direction  (§  192). 

cos  <p  tan  6  —  sin  <p  cos  /' 

See  §  193. 
Curvature  correction 

I  (2  sin'-Jz//,       2  sin'£j/a  2  sin' 

"     +  '  •  '  " 


sn  i 

in  which  At^  At»  .  .  .  Atn  are  the  differences  between  each 
hour-angle  and  the  mean  of  the  hour-angles,  and  the  values  of 

the  terms  —  :  —  ^77-*,  etc.,  are  known  from  the  table  in  §  307. 
sin  i 

(See  §§  194-197.) 

Correction  for  diurnal  aberration, 

cos  0  cos  z 


Cos 


or,  with  sufficient  accuracy  for  nearly  all  cases  0^.32.  See 
§§  198,  199.  The  sign  of  the  correction  is  -f-  if  applied  to 
the  computed  azimuth  of  the  star  expressed  as  an  angle  east 


342  GEODETIC  ASTRONOMY.  §31 1. 

of  north,  and  —  if  the  angle  is  measured  westward  from  the 
north. 

To  compute  azimuth  from  observations  upon  a  circumpolar 
star  with  a  repeating  instrument. 

Example  of  record,  §  203. 

The  computation  is  made  as  indicated  above  with  the 
exception  of  certain  modifications  indicated  in  §  204. 

To  compute  azimuth  from  observations  upon  a  circumpolar 
star  with  an  eyepiece  micrometer. 

See  example  of  record  and  computation,  §§210-214. 


§312. 


LIST   OF  DEFINITIONS. 


343 


312. 


LIST  OF  DEFINITIONS. 


Aberration 41 

Aberration,  annual 41 

Aberration,  diurnal 41 

Aberration,  planetary 41 

Absolute  Personal  Equation....  144 

Accidental  Error 86 

Altitude 14 

Annual  Aberration 41 

Apparent  Place 41 

Apparent  Solar  Day 19 

Apparent  Solar  Time 21 

Argument 22 

Artificial  Horizon 65 

Astronomical  Day 22 

Astronomical  Refraction 76 

Azimuth 14 

Azimuth,  astronomical 16 

Azimuth,  geodetic 16 

Besselian  Fictitious  Year 43 

Celestial  Sphere 10 

Civil  Day 22 

Collimation,  error  of 100 

Collimation,  line  of 98 

Constant  Error 86 

Culminate 150 

Culmination 18 

Day 18,  19,  21,  22 

Declination 12 

Diurnal  Aberration 41 

Ecliptic 5,  10,  ii 

Ecliptic,  mean 43 

Equation  of  Time 22 

Equator 10,  n 


Equatorial  Horizontal  Parallax.  74 

Equatorial  Intervals 108 

Equator,  mean 43 

Equinox,  vernal,  autumnal 12 

Errors 82,  86 

Error,  index 64 

Error  of  Collimation 100 

Error  of  Interpolation 32 

Errors  of  Observation 82 

Error  of  Runs 207 

Explement 220 

External  Errors 82 

Fictitious  Year  (Besselian) 43 

Horizon n 

Horizon,  artificial 65 

Horizon-glass 59 

Horizontal  Parallax 74 

Hour-angle 13 

Hour-circle 11 

Independent  Star-numbers 51 

Index  Correction 64 

Index  Error 64 

Index-glass 60 

Instrumental  Errors 82 

Interpolation 31 

Intervals,  equatorial 108 

Latitude,  astronomical 14,  15 

Latitude,  celestial 15 

Latitude,  geodetic 16 

Limb 69 

Line  of  Collimation 98 

Longitude,  astronomical 14,  15 

Longitude,  celestial 15 


344 


GEODETIC  ASTRONOMY. 


§312. 


Longitude,  geodetic 16 

Lunar  Distance 263 

Mean  Ecliptic 43 

Mean  Equator 43 

Mean  Place 41,  42,  43 

Mean  Refraction 77 

Mean  Solar  Day 21 

Mean  Solar  Time 21 

Mean  Sun 21 

Mean  Time 21 

Meridian 12 

Meridian  Line 13 

Meridian  Plane 12 

Nadir 12 

Nutation 5,  6 

Observer's  Errors 82 

Optical  Centre 152 

Parallax 74 

Parallax,  sextant 62 

Personal  Equation 144 

Planetary  Aberration 41 

Pole..  ii 


PAGB 

Polar  Distance 8 

Precession 5 

Prime  Vertical 13 

Probable  Error 272-274 

Proper  Motion 48 

Refraction 76 

Refraction,  mean 77 

Relative  Personal  Equation....  144 

Right  Ascension 12 

Run  (of  a  micrometer) 207 

Sextant  Parallax 62 

Sidereal  Day 18 

Sidereal  Time 18 

Standard  Time 23 

Star-numbers,  Independent 51 

Station  Error 15 

Transit 18 

True  Place 41 

Vertical  Circle 14 

Zenith 12 

Zenith  Distance 14 


INDEX. 


PAGES 

A,  B,  C,  factors 296-312 

Aberration,  correction,  time 3Ia 

Aberration,  diurnal,  annual,  and  planetary 41-42 

correction  for,  with  transit  observations 116-118 

correction  for,  in  azimuth  computations 216-218 

Adjustments,  of  altazimuth 200-201 

of  sextant 62-65 

of  transit 99-103 

of  zenith  telescope 153-156 

Altazimuth,  description  of 197-200 

adjustments  of 200-201 

Altitude,  defined !4 

of  Polaris 320 

Apparent  places,  computation  of 50-54 

Apparent  solar  day ig 

Apparent  solar  time 19-21 

Astronomical  day 22 

Azimuth 197-240 

Azimuth  adjustment  of  transit 101-103 

Azimuth,  astronomical  versus  geodetic 16 

Azimuth  correction,  time  computation 117-118 

Azimuth,  curvature  correction  in  computation  of 213-215 

defined 14 

directions  for  observing 201-205,  220,  221,  223-225 

discussion  of  errors 230-233 

diurnal  aberration  correction 216-218 

example  of  record 205 ,  222-225 

formula 211-213 

method  of  repetitions 220-223 

micrometric 223-228 

observations  upon  Sun 234-238 

345 


INDEX. 

PAGES 

Azimuth,  of  Polaris 322-327 

questions  and  examples 238-240 

with  an  astronomical  transit 233-234 

with  an  engineer's  transit 234-238 

Barometric  correction,  refraction 293 

Celestial  sphere,  denned IO 

Chronograph 105-107 

Chronometric  longitudes 253-262 

Circle  reading 206-209 

Civil  day 22 

Collimation,  correction,  transit  observations 118-119 

line  of,  denned 98 

Combination  of  results,  zenith  telescope 168-174 

Computation,  of  apparent  places 40,  50-54 

of  azimuth 218-219,  225-226,  237 

of  chronometric  longitudes , 256-262 

of  mean  places 43-48 

of  observations  with  zenith  telescope 166-168 

of  telegraphic  longitudes 244-247 

of  time  from  sextant  observations 71-72 

of  transit  time  observations,  without  least  squares, 

107,  120-125 

of  transit  time  observations,  by  least  squares. ..107,  126-133 

Computing,  suggestions  about 270-272 

Conversion,  mean  to  sidereal  time 24-25,  279-282 

sidereal  time  to  mean  solar 26,  283-286 

Counting  seconds 68-69 

Covarrubias,  method  of  observing  with  sextant 93~94 

Culmination,  defined 18 

Curvature  correction,  azimuth 213-215 

micrometer  computation 315 

Day,  sidereal 18 

apparent  solar 19 

mean  solar 21 

civil  and  astronomical 22 

Declination,   denned 11-12 

Definitions,  list  of 343-344 

Differential  refraction  correction,  latitude 314 

Dip  of  sea  horizon 295 


INDEX.  347 

PAGES 

Diurnal  aberration  correction,  azimuth 216-218 

time 116-118,  313 

Eccentricity  of  sextant 85,  88-90 

Ecliptic 10-11 

Economics  of  observing 276-278 

of  zenith  telescope  observations 186-187 

Ephemeris 26,  38-40 

Equation  of  time 22 

Equation,  personal 249-252 

Equator 10-11 

Equatorial  intervals 133-134 

Equinoxes 12 

Errors,  constant  versus  accidental 86-87 

in  azimuth  observations 230-233 

in  sextant  observations 82-88 

probable 272-274 

station 276-278 

of  telegraphic  longitudes 248-249,  251 

with  transit 140-144 

with  zenith  telescope 181-187 

Examples 27-29,  55-58,  94-95,  146-149,  194-196,  238-240 

Eye  and  ear  observations  of  time 68-69,   103-104 

Factors,  used  in  time  computations 296-312 

Fictitious  year 43 

Formulae  and  notation 328-342 

Horizon,  defined ir 

dip  of 295 

Hour-angle 13 

Hour-circles n 

Index  error  of  sextant 64-65,  70 

Inequality  of  pivots 134 

Interpolation 3i~39 

Interpolation,  accuracy  of,  for  Sun  and  Planets 38-39 

accuracy  of,  for  Moon 39 

along  a  chord 32-33 

along  a  parabola 35~38 

along  a  tangent 33~34 

Jupiter  observed  for  longitude 267 


34&  INDEX. 

PAGES 

Latitude,  astronomical 14 

celestial !g 

from  altitudes  of  circumpolars 191-193 

from  altitude  of  Polaris 320 

from  circummeridian  zenith  distances 189-191 

from  observed  altitudes 188 

from  zenith  telescope  observations 150-187 

geodetic 15-16 

variation 4,  274-276 

Level  value,  by  comparison  with  another  level 136-137 

with  eyepiece  micrometer 136 

with  level-tester 135-136 

with   theodolite 137-139 

List,  observing,  for  zenith  telescope 156-160 

of  definitions 343~344 

Longitude,  astronomical 14 

by  chronometers 253-262 

by  flashes 252-253 

by  Moon  observations 262-267 

by  observations  upon  Jupiter 267 

by  telegraph 242-249 

Lunar  distances 263-267 

2  sin5  \t 


Mean  line,  reduction  to 108-112 

Mean  place,  defined 42-43 

reduction  of,  from  year  to  year 43-48 

Mean  refraction 292-293 

Mean  solar  to  sidereal  time 24-25,  279-282 

Mean   Sun 21 

Meridian,  defined 12-13 

reduction  to 315 

Micrometer,  correction  for  run,  table 321 

Micrometer  value 174-181 ,  228-230 

Micrometer  value,  curvature  correction 315 

Micrometric  method,  azimuth 223-228 

Microscopes,  reading 198-199 

Moon,  motion  of 6 

observed  for  longitude 262-267 

position  of 30 

2  sin4^/ 

.319 


INDEX.  349 

PAGES 

North  polar  distance  ...........................  ;  ................  5 

Notation  and  formulae  ...........................................   328-342 

Nutation  .......................................................  _ 


_ 

'-~^f'                          .......................................  319 

Observing,  economics  of  ........................................  276-278 

Observing  list  for  zenith  telescope  ..............................  156-160 

Observing,  suggestions  about  ...................................  268-270 

Occultations,  observed  for  longitude  .............................  264-267 

Parallax  ......................................................  73~75,  291 

Personal  equation  ..............................................   249-252 

Pivot  inequality  ...................    ............................  134 

Planets,  position  of  ............................................  31 

Polar  distance  ..................................................  8 

Polaris,  altitude  of  ..............................................  320 

azimuth  of  ....................  «  .........................   322-327 

Poles  ..........................................................  n 

Positions  of  Sun,  Moon,  and  stars  ..............................       39~4i 

Precession,  change  of  annual  ...............    ...................   287-290 

defined  ..........................  ...................  5-6 

Prime  vertical  ..................................................  13 

Probable  errors    ................................................   272-274 

Proper  motion  ..................................................       48-50 

Questions  ....................  27-29,  55~58,  94~95»  146-149,  194-196,  238-240 

Rate  correction,  time  observations  ...............................    119-120 

Reading  microscopes  ............................................    198-199 

Refraction  ................................................  75~8o,  292-294 

Refraction  correction,  zenith  telescope  .......................  164-165,  314 

Relative  weights,  transit  observations  .......................  131-133,  313 

Repetitions,  method  of,  in  observing  azimuth  ....................   220-223 

Right  ascension,  computation  of  ................................       30-31 

defined  ........................................       12-13 

Run  of  a  micrometer,  table  of  ...................................  321 

Sea  horizon,  dip  of  ..............................................  295 

Secular  variation  in  precession  ..................................  287-290 

Sextant,  adjustments  of  .........................................  62-65 

computation  of  time  ...................................  71-72 

correction  for  eccentricity  ..............................  88-90 


350  INDEX. 

PAGES 

Sextant,  description  of 59-60 

errors 82-88 

observations  at  equal  altitudes 92-93 

observations  at  sea 91-92 

principle  of 60-62 

record 71 

Sidereal  day 18 

Sidereal  time,  defined 18-19 

converted  to   mean  solar 26,  283-286 

Solar  time,  apparent •. 19-22 

mean 21-22 

Standard  time 23-24 

Stars,  apparent  motion  of. 8-10 

distance  to 7-8 

positions  of 39~4i 

Station  errors 15-16,  276-278 

Suggestions,  about  computing 270-272 

about  observing 268-270 

Sumner's  method 9r-92 

Sun,  mean 21 

position  of 31,  38-39 

parallax  of 291 

Telegraphic  longitudes 242-249 

Temperature  correction,  refraction 294 

Time,  apparent  solar 19-21 

by  eye  and  ear 68-69,   103-104 

equation  of 22 

factors c    296-312 

mean  solar 4 21-22 

sidereal 18-19 

standard 23-24 

Transit,  adjustments  of 99-103 

defined 18 

description  of 96-98 

directions  for  observing  with 103-105 

errors  140-144 

in  prime  vertical 145,  146 

in  the  vertical  of  Polaris 145 

out  of  the  meridian 145 

record 107 

theory  of 98-99 


INDEX.  351 

PAGES 

Unequal  weights,  transit  observations 131-133 

Variation  of  latitude 274-276 

Vertical  circle,  defined 14 

Weights,  transit  observation 131-133,  313 

for  different  chronometers 260-262 

Zenith 12 

Zenith  distance 14 

Zenith  telescope,  adjustments  of 153-156 

computation 166-168 

derivation  of  formula 163-166 

description  of 151-153 

directions  for  observing 160-162 

errors 181-187 

observing  list  for 156-160 

principle  of 150-151 

record 162 


"*»* 

OF   THK 

UNIVERSITY 


*F 


FIG.  I 

CELESTIAL  SPHERE. — C  =  position  of  observer  ;  Z  =  his  zenith  and  N 
his  nadir  ;  P,  P'  —  the  north  and  south  poles  ;  HMEOW—  horizon  ;  ZHf, 
ZH,  ZO  —  vertical  circles  ;  O,  E,  H,  W  —  north,  east,  south,  and  west 
points,  respectively  ;  ALEQW  =  equator  ;  S  =  a  star  ;  S'SD  =  small 
circle  defining  the  diurnal  motion  of  the  star  ;  PAP',  PSP'  =  hour-circles; 
OPZAH  =  the  meridian  ;  ZA  —  latitude  of  station  ;  ZS  =  zenith  distance 
of  star  ;  PS  =  north  polar  distance  of  star  ;  LS  =  declination  of  star  ; 
ZPS  =  hour-angle  of  star  ;  MS  =  altitude  of  star  ;  OM=  azimuth  of  star. 


FIG.  la 

CELESTIAL  SPHERE. — P,  P1  =  north  and  south  poles  ;  EVQA  =  equator  ; 
VD AB  =  ecliptic  (the  arrows  indicating  the  apparent  motion  of  the  Sun) ; 
V  =  vernal  equinox  ;  A  =  autumnal  equinox  ;  VL  =  right  ascension  of 
the  star  S. 


SEXTANT 


VERNAL 
EQUINOX 


FIG.  2 


EYE 


^     OK  THB1 

UNIVERSITY 
OF 


FIG.  5 


EYE 


FIG.  6 


FIG.  7 


POLE 


FIG  8 


ASTRONOMICAL  TRANSIT, 


MERIDIAN  TELESCOPE, 


A 
1*5 


LINE  I 
FIG.  12 


B 
2*0 


FIG.  16 


FIG.  15 


FIG.  17 


T^L™"1*^ 

UNIVERSITY 


ZENITH  TELESCOPE. 


FIG.  21 


FIG.  22 


FIG.  24.-ALTAZIMUTH. 


OF  THK 

UNIVERSITY 


IICROMETER  LINES 
FIG.  25 


A     D 


FIG.  27 


OFTHB 

UNIVERSITY 

f 


LOCAL  BATTERY 


CHRONOMETER 


CHRONOGRAPH 


OBSERVING  KEYS  (CLOSED) 


TELEGRAPHERS  KEY 

MAIN    LINE 


SOUNDER  RELAY 


SOUNDER 


LOCAL  BATTERY 


FIG.  30 


MAIN    LINE 


LOCAL  BATTERY 


CHRONOMETER 


OBSERVING  KEYS  (CLOSED) 


'SOUNDER  RELAY 
LOCAL  BATTERY 


MAIN  LINE 
TELEGRAPHERS  KEY 


MAIN  LINE: 


SOUNDER 


FIG- 31 


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Rifle,  Caliber  .45 18mo,  paper,  15 

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Powell's  Army  Officer's  Examiner 12mo,  4  00 

Reed's  Signal  Service , 50 

Sharpe's  Subsisting  Armies 18mo,  morocco,  1  50 

Very's  Navies  of  the  World 8vo,  half  morocco,  3  50 

Wheeler's  Siege  Operations 8vo,  2  00 

Wmthrop's  Abridgment  of  Military  Law 12mo,  2  50 

Woodhull's  Notes  on  Military  Hygiene 12mo,  1  50 

Young's  Simple  Elements  of  Navigation..  12mo,  morocco  flaps,  2  00 

first  edition 1  00 

ASSAYING. 

SMELTING — ORE  DRESSING— ALLOYS,  ETC. 

Fletcher's  Quant.  Assaying  with  the  Blowpipe..  12mo,  morocco,  1  50 

Furman's  Practical  Assaying 8vo,  3  00 

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*  Mitchell's  Practical  Assaying,     (Crookes.) 8vo,  10  00 

O'Driscoll's  Treatment  of  Gold  Ores 8vo,  2  CO 

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Wilson's  Cyanide  Processes 12mo,  1  50 

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Craig's  Azimuth 4to,  3  50 

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Gore's  Elements  of  Geodesy 8vo,  2  50 

3 


Hay  ford's  Text-book  of  Geodetic  Astronomy 8vo. 

Michie  and  Harlow's  Practical  Astronomy 8vo,     $3  00 

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Thome's  Structural  Botany 18mo,  2  25 

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BRIDGES,  ROOFS,   Etc. 

CANTILEVER — DRAW — HIGHWAY — SUSPENSION. 
(See  also  ENGINEERING,  p.  6.) 

Boiler's  Highway  Bridges 8vo,  2  00 

*  "  The  Thames  River  Bridge 4to,  paper,  5  00 

Burr's  Stresses  in  Bridges. . . ,  8vo,  3  50 

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Dredge's  Thames  Bridges 7  parts,  per  part,  1  25 

Du  Bois's  Stresses  in  Framed  Structures 4to,  10  00 

Foster's  Wooden  Trestle  Bridges 4to,  5  00 

Greene's  Arches  in  Wood,  etc 8vo,  2  50 

"  Bridge  Trusses 8vo,  250 

Roof  Trusses 8vo,  125 

Howe's  Treatise  on  Arches 8vo,  4  00 

Johnson's  Modern  Framed  Structures 4to,  10  00 

Merriman  &  Jacoby's  Text-book  of  Roofs  and  Bridges. 

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Merriman  &  Jacoby's  Text-book  of  Roofs  and  Bridges. 

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Merriman  &  Jacoby's  Text-book  of  Roofs  and  Bridges. 

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Part  IV.,  Continuous,  Draw,  Cantilever,  Suspension,  and 

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*Morisou's  The  Memphis  Bridge Oblong  4to,  10  00 

4 


Waddell's  Iron  Highway  Bridges 8vo,  f  4  00 

De  Pontibus  (a  Pocket-book  for  Bridge  Engineers). 

Wood's  Construction  of  Bridges  and  Roofs 8vo,  2  00 

Wright's  Designing  of  Draw  Spans 8vo,  2  50 

CHEMISTRY. 

QUALITATIVE — QUANTITATIVE-  ORGANIC — INORGANIC,  ETC. 

Adriauce's  Laboratory  Calculations 12mo,  1  25 

Allen's  Tables  for  Iron  Analysis 8vo,  3  00 

Austen's  Notes  for  Chemical  Students 12mo,  1  50 

Bolton's  Student's  Guide  in  Quantitative  Analysis 8vo,  1  50 

Classen's  Analysis  by  Electrolysis.   (Herrick  and  Boltwood.).8vo,  3  00 

Crafts's  Qualitative  Analysis.     (Schaeffer.) 12rno,  1  50 

Drechsel's  Chemical  Reactions.    (Merrill.) 12ino,  1  25 

Fresenius's  Quantitative  Chemical  Analysis.    (Allen.) 8vo,  6  00 

"          Qualitative          "              "            (Johnson.) 8vo,  300 

(Wells)  Trans.  16th. 

German  Edition '.8vo,  5  00 

Fuerte's  Water  and  Public  Health 12mo,  1  50 

Gill's  Gas  and  Fuel  Analysis 12mo,  1  25 

Harnmarsten's  Physiological  Chemistry.    (Mandel.) 8vo,  4  00 

Helm's  Principles  of  Mathematical  Chemistry.    (Morgan).  12mo,  1  50 

Kolbe's  Inorganic  Chemistry 12mo,  1  50 

Ladd's  Quantitative  Chemical  Analysis 12mo,  1  00 

Lanclauer's  Spectrum  Analysis.     (Tingle.) 8vo,  3  00 

Handel's  Bio-chemical  Laboratory 12mo,  1  50 

,..         .    T_r                 ,                                                                    _  _  ... 

Mason  s  Water-supply 8vo,  o  00 

"      Analysis  of  Potable  Water.     (In  the  press.} 

Miller's  Chemical  Physics 8vo,  2  00 

Mixter's  Elementary  Text-book  of  Chemistry 12mo,  1  50 

Morgan's  The  Theory  of  Solutions  and  its  Results 12mo,  1  00 

Nichols's  Water-supply  (Chemical  and  Sanitary) 8vo,  2  50 

O'Brine's  Laboratory  Guide  to  Chemical  Analysis 8vo,  2  00 

Perkins's  Qualitative  Analysis 12mo,  1  00 

Pinner's  Organic  Chemistry.     (Austen.) 12mo,  1  50 

Poole's  Calorific  Power  of  Fuels 8vo,  3  00 

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5 


Ruddiinau's  Incompatibilities  in  Prescriptions 8vo,  $2  00 

Sehimpf  s  Volumetric  Analysis 12mo,  2  50 

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ELEMENTARY — GEOMETRICAL — TOPOGRAPHICAL. 

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MacCord's  Descriptive  Geometry 8vo,  3  00 

"          Kinematics 8vo,  500 

"           Mechanical  Drawing 8vo,  400 

Mahan's  Industrial  Drawing.    (Thompson.) 2  vols.,  8vo,  3  50 

Reed's  Topographical  Drawing.     (II.  A.) 4to,  5  00 

Reid's  A  Course  in  Mechanical  Drawing 8vo.  2  00 

"      Mechanical  Drawing  and  Elementary  Machine  Design. 

8vo. 

Smith's  Topographical  Drawing.     (Macmillan.) 8vo,  2  50 

Warren's  Descriptive  Geometry 2  vols.,  8vo,  3  50 

"        Drafting  Instruments 12mo,  1  25 

' '        Free-hand  Drawing 1 2nio,  1  00 

"        Higher  Linear  Perspective  8vo,  3  50 

"        Linear  Perspective 12mo,  1  00 

"        Machine  Construction 2  vols. ,  8vo,  7  50 

"        Plane  Problems , 12mo,  125 

"         Primary  Geometry 12mo,  75 

"        Problems  and  Theorems 8vo,  250 

"         Projection  Drawing 12mo,  150 

Shades  and  Shadows 8vo,  300 

"        Slereotomy— Stone  Cutting 8vo,  250 

Whelpley's  Letter  Engraving 12mo,  2  00 

6 


ELECTRICITY  AND  MAGNETISM. 

ILLUMINATION— BATTERIES— PHYSICS. 

Anthony  and  Brackett's  Text-book  of  Physics  (Magie).   . .  .8vo,  $4  00 

Barker's  Deep-sea  Soundings 8vo,  2  00 

Benjamin's  Voltaic  Cell -. 8vo,  3  00 

History  of  Electricity 8vo  3  00 

Cosmic  Law  of  Thermal  Repulsion 18mo,  75 

Crehore  and  Squier's  Experiments  with  a  New  Polarizing  Photo- 
Chronograph 8vo,  3  00 

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Vol.11 4to,  750 

Gilbert's  De  magnete.     (Mottelay.) 8vo,  2  50 

Holman's  Precision  of  Measurements 8vo,  2  00 

Michie's  Wave  Motion  Relating  to  Sound  and  Light 8vo,  4  00 

Morgan's  The  Theory  of  Solutions  and  its  Results 12nio,  1  00 

Niaudet's  Electric  Batteries.     (Fishback.) .  .12mo,  2  50 

Reagan's  Steam  and  Electrical  Locomotives 12mo,  2  00 

Thurston's  Stationary  Steam  Engines  for  Electric  Lighting  Pur- 
poses  12mo,  150 

Tillman's  Heat 8vo,  1  50 

ENGINEERING. 

CIVIL — MECHANICAL— SANITARY,  ETC. 
(See  also  BRIDGES,   p.  4 ;  HYDRAULICS,   p.  8 ;  MATERIALS  OF  EN- 

GINEERING,  p.  9  ;  MECHANICS  AND  MACHINERY,  p.  11  ;    STEAM  ENGINES 

AND  BOILERS,  p.  14.) 

Baker's  Masonry  Construction 8vo,  5  00 

Surveying  Instruments 12mo,  300 

Black's  U.  S.  Public  Works 4to,  5  00 

Brook's  Street  Railway  Location 12mo,  morocco,  1  50 

Butts's  Engineer's  Field-book 12rno,  morocco,  2  50 

Byrne's  Highway  Construction 8vo,  7  50 

"        Inspection  of  Materials  and  Workmanship.   12mo,  mor. 

Carpenter's  Experimental  Engineering 8vo,  6  00 

Church's  Mechanics  of  Engineering — Solids  and  Fluids 8vo,  6  00 

"        Notes  and  Examples  in  Mechanics 8vo,  200 

Crandall's  Earthwork  Tables  , 8vo,  1  50 

The  Transition  Curve 12mo,  morocco,  1  50 

7 


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*  Drinker's  Tunnelling 4to,  half  morocco,  25  00 

Eissler's  Explosives — Nitroglycerine  and  Dynamite 8vo,  4  00 

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Gore's  Elements  of  Geodesy : 8vo,  2  50 

Howard's  Transition  Curve  Field-book. , .  .  .12mo,  morocco  flap,  1  50 

Howe's  Retaining  Walls  (New  Edition.) , 12mo,  1  25 

Hudson's  Excavation  Tables.     Vol.  II 8vo,  1  00 

Button's  Mechanical  Engineering  of  Power  Plants 8vo,  5  00 

Johnson's  Materials  of  Construction 8vo,  6  00 

Stadia  Reduction  Diagram.  .Sheet,  22£  X  28i  inches,  50 

"         Theory  and  Practice  of  Surveying 8vo,  4  00 

Kent's  Mechanical  Engineer's  Pocket-book 12mo,  morocco,  5  00 

Kiersted's  Sewage  Disposal 12nio,  1  25 

Kirkwood's  Lead  Pipe  for  Service  Pipe 8vo,  1  50 

Mahan's  Civil  Engineering.      (Wood.) 8vo,  5  00 

"Merrinian  and  Brook's  Handbook  for  Surveyors. . .  .12mo,  mor.,  2  00 

Merriman's  Geodetic  Surveying 8vo,  2  00 

"          Retaining  Walls  and  Masonry  Dams 8vo,  2  00 

Mosely's  Mechanical  Engineering.     (Mahau.) Svo,  5  00 

Nagle's  Manual  for  Railroad  Engineers .  .12ino,  morocco,  8  00 

Patton's  Civil  Engineering Svo,  7  50 

"       Foundations 8vo,  5  00 

Hockwell's  Roads  and  Pavements  in  France 12mo,  1  25 

Ruffuer's  Non-tidal  Rivers  Svo,  1  25 

Searles's  Field  Engineering 12mo,  morocco  flaps,  3  00 

Railroad  Spiral 12mo,  morocco  flaps.  1  50 

Siebert  and  Biggin's  Modern  Stone  Cutting  and  Masonry. .  .8vo,  1  50 

Smith's  Cable  Tramways 4to,  2  50 

Wire  Manufacture  and  Uses 4to,  3  00 

Spalding's  Roads  and  Pavements 12mo,  2  00 

Hydraulic  Cement 12mo,  200 

Thurstou's  Materials  of  Construction .8vo,  5  00 

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"          Cross-section Sheet,  25 

*'           Excavations  and  Embankments Svo,  2  00 

S 


*  Trautwiue's  Laying  Out  Curves 12mo,  morocco,  $2  50 

Waddell's  De  Pontibus  (A  Pocket-book  for  Bridge  Engineers;. 

12mo,  morocco,  3  00 

Wait's  Engineering  and  Architectural  Jurisprudence 8vo,  6  00 

Sheep,  6  50 

Law  of  Field  Operation  in  Engineering,  etc 8vo. 

Warren's  Stereotomy— Stone  Cutting 8vo,  2  50 

Webb  s  Engineering  Instruments 12mo,  morocco,  1  00 

Wegmauu's  Construction  of  Masonry  Dams 4to,  5  00 

Wellington's  Location  of  Railways. 8vo,  5  00 

Wheeler's  Civil  Engineering 8vo,  4  00 

Wolff's  Windmill  as  a  Prime  Mover 8vo,  3  00 

HYDRAULICS. 

WATER-WHEELS — WINDMILLS — SERVICE  PIPE — DRAINAGE,  ETC. 

(See  also  ENGINEERING,  p.  6.) 
Bazin's  Experiments  upon  the  Contraction  of  the  Liquid  Vein 

(Trautwiue) 8vo,  2  00 

Bovey's  Treatise  on  Hydraulics 8vo,  4  00 

Coffin's  Graphical  Solution  of  Hydraulic  Problems 12mo,  2  50 

Fen-el's  Treatise  on  the  Winds,  Cyclones,  and  Tornadoes. .  .8vo,  4  00 

Fuerte's  Water  and  Public  Health 12mo,  1  50 

Ganguillet  &  Kutter'sFlow  of  Water.  (Hering&  Trautwine.).8vo,  4  00 

Hazen's  Filtration  of  Public  Water  Supply 8vo,  2  00 

Herschel's  115  Experiments  8vo,  2  00 

Kiersted's  Sewage  Disposal 12mo,  1  25 

Kirkwood's  Lead  Pipe  for  Service  Pipe  .  „ 8vo,  1  50 

Mason's  Water  Supply 8vo,  5  00 

Merrirnaii's  Treatise  on  Hydraulics. . , 8vo,  4  00 

Nichols's  Water  Supply  (Chemical  and  Sanitary) 8vo,  2  50 

Ruffner's  Improvement  for  Non-tidal  Rivers 8vo,  1  25 

Wegmaun's  Water  Supply  of  the  City  of  New  York 4to,  10  00 

Weisbach's  Hydraulics.     (Du  Bois.) 8vo,  5  00 

Wilson's  Irrigation  Engineering 8vo,  400 

Hydraulic  and  Placer  Mining 12mo,  2  00 

Wolff's  Windmill  as  a  Prime  Mover 8vo,  3  00 

Wood's  Theory  of  Turbines 8vo,  2  50 

MANUFACTURES. 

ANILINE — BOILERS— EXPLOSIVES— IRON— SUGAR — WATCHES  — 
WOOLLENS,  ETC. 

Allen's  Tables  for  Iron  Analysis 8vo,  3  00 

Beaumont's  Woollen  and  Worsted  Manufacture 12ino,  1  50 

Bollaud's  Encyclopaedia  of  Founding  Terms 12mo,  3  00 

9 


Bolland's  The  Iron  Founder 12mo,  2  50 

"        Supplement 12mo,  2  50 

Booth's  Clock  and  Watch  Maker's  Manual 12mo,  2  00 

Bouvier's  Handbook  on  Oil  Painting 12mo,  2  00 

Eissler's  Explosives,  Nitroglycerine  and  Dynamite 8vo,  4  00 

Ford's  Boiler  Making  for  Boiler  Makers 18mo,  1  00 

Metcalfe's  Cost  of  Manufactures 8vo,  5  00 

Metcalf 's  Steel— A  Manual  for  Steel  Users 12mo,  $2  00 

Reimann's  Aniline  Colors.     (Crookes.) 8vo,  2  50 

*  Reisig's  Guide  to  Piece  Dyeing 8vo,  25  00 

Spencer's  Sugar  Manufacturer's  Handbook 12mo,  inor.  flap,  2  00 

"        Handbook      for      Chemists      of      Beet       Houses. 

12mo,  mor.  flap,  3  00 

Svedelius's  Handbook  for  Charcoal  Burners 12mo,  1  50 

The  Lathe  and  Its  Uses .8vo,  6  00 

Thurston's  Manual  of  Steam  Boilers 8vo,  5  00 

Walke's  Lectures  on  Explosives 8vo,  4  00 

West's  American  Foundry  Practice , 12mo,  2  50 

"      Moulder's  Text-book 12mo,  2  50 

Wiechmanu's  Sugar  Analysis 8vo,  2  50 

Woodbury's  Fire  Protection  of  Mills 8vo,  2  50 


MATERIALS  OF  ENGINEERING. 

STRENGTH — ELASTICITY — RESISTANCE,  ETC. 
(See  also  ENGINEERING,  p.  6.) 

Baker's  Masonry  Construction 8vo,  5  00 

Beardslee  and  Kent's  Strength  of  Wrought  Iron 8vo,  1  50 

Bovey's  Strength  of  Materials 8vo,  7  50 

Burr's  Elasticity  and  Resistance  of  Materials 8vo,  5  00 

Byrne's  Highway  Construction 8vo,  5  00 

Carpenter's  Testing  Machines  and  Methods  of  Testing  Materials. 

Church's  Mechanics  of  Engineering — Solids  and  Fluids 8vo,  6  00 

Du  Bois's  Stresses  in  Framed  Structures 4to,  10  00 

Hatfield's  Transverse  Strains 8vo,  5  00 

Johnson's  Materials  of  Construction 8vo,  6  00 

Lanza's  Applied  Mechanics 8vo,  7  50 

Merrill's  Stones  for  Building  and  Decoration 8vo,  5  00 

Merriman's  Mechanics  of  Materials ; . . .  .8vo,  4  00 

Strength  of  Materials 12rno,  1  00 

Pattou's  Treatise  on  Foundations 8vo,  5  00 

Rockwell's  Roads  and  Pavements  in  France 12mo,  1  25 

Spaldiug's  Roads  and  Pavements 12mo,  2  00 

Thurston's  Materials  of  Construction , 8vo,  5  00 

10 


Thurston's  Materials  of  Engineering 3  vols.,  8vo,  $8  00 

Vol.  I. ,  Non-metallic  8vo,  2  00 

Vol.  II.,  Iron  and  Steel 8vo,  3  50 

Vol.  III.,  Alloys,  Brasses,  and  Bronzes 8vo,  2  50 

Weyrauch's  Strength  of  Iron  and  Steel.    (Du  Bois.) 8vo,  1  50 

Wood's  Resistance  of  Materials 8vo,  2  00 

MATHEMATICS. 

CALCULUS— GEOMETRY— TRIGONOMETRY,  ETC. 

Baker's  Elliptic  Functions 8vo,  1  50 

Ballard's  Pyramid  Problem . . . .8vo,  1  50 

Barnard's  Pyramid  Problem 8vo,  1  50 

Bass's  Differential  Calculus 12mo,  4  00 

Brigg's  Plane  Analytical  Geometry 12mo,  1  00 

Chapman's  Theory  of  Equations 12mo,  1  50 

Chessin's  Elements  of  the  Theory  of  Functions. 

Compton's  Logarithmic  Computations 12mo,  1  50 

Craig's  Linear  Differential  Equations 8vo,  5  00 

Davis's  Introduction  to  the  Logic  of  Algebra 8vo,  1  50 

Halsted's  Elements  of  Geometry ,..8vo,  1  75 

"       Synthetic  Geometry 8vo,  1  50 

Johnson's  Curve  Tracing 12mo,  1  00 

Differential  Equations— Ordinary  and  Partial 8vo,  350 

Integral  Calculus 12mo,  150 

"  "  "         Unabridged. 

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Ludlow's  Logarithmic  and  Other  Tables.     (Bass.). . .' 8vo,  2  00 

Trigonometry  with  Tables.     (Bass.) 8vo,  300 

Mahan's  Descriptive  Geometry  (Stone  Cutting). ...   8vo,  1  50 

Merriman  and  Woodward's  Higher  Mathematics 8vo,  5  00 

Merrimau's  Method  of  Least  Squares 8vo,  2  00 

Parkers  Quadrature  of  the  Circle , 8vo,  2  50 

Rice  and  Johnson's  Differential  and  Integral  Calculus, 

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"                 Differential  Calculus 8vo,  300 

Abridgment  of  Differential  Calculus.... 8vo,  150 

Searles's  Elements  of  Geometry 8vo,  1  50 

Totten's  Metrology 8vo,  2  50 

Warren's  Descriptive  Geometry 2  vols.,  8vo,  3  50 

"        Drafting  Instruments 12mo,  125 

"        .Free-hand  Drawing 12mo,  100 

"        Higher  Linear  Perspective 8vo,  3  50 

"        Linear  Perspective ,12mo,  1  00 

"        Primary  Geometry 12mo,  75 

11 


Warren's  Plane  Problems 12mo,  $1  25 

"        Problems  and  Theorems 8vo,  2  50 

Projection  Drawing 12mo,  1  50 

Wood's  Co-ordinate  Geometry 8vo,  2  00 

Trigonometry 12mo,  1  00 

Woolf's  Descriptive  Geometry Royal  8vo,  3  00 

MECHANICS-MACHINERY. 

TEXT-BOOKS  AND  PRACTICAL  WORKS. 
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Baldwin's  Steam  Heating  for  Buildings 12mo,  2  50 

Benjamin's  Wrinkles  and  Recipes 12mo,  2  00 

Carpenter's  Testing  Machines  and   Methods  of  Testing 

Materials 8vo. 

Chordal's  Letters  to  Mechanics 12mo,  2  00 

Church's  Mechanics  of  Engineering 8vo,  6  00 

"        Notes  and  Examples  in  Mechanics 870,  2  00 

Crehore's  Mechanics  of  the  Girder 8vo,  5  00 

Cromwell's  Belts  and  Pulleys 12mo,  1  50 

Toothed  Gearing 12mo,  150 

Compton's  First  Lessons  in  Metal  Working 12mo,  1  50 

Dana's  Elementary  Mechanics 12mo,  1  50 

Dingey's  Machinery  Pattern  Making 12mo,  2  00 

Dredge's     Trans.     Exhibits     Building,      World     Exposition, 

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Du  Bois's  Mechanics.     Vol.  I.,  Kinematics 8vo,  3  50 

"               "               Vol.11..  Statics 8vo,  400 

Vol   III.,  Kinetics 8vo,  3  50 

Fitzgerald's  Boston  Machinist 18nio,  1  00 

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Jones  Machine  Design.     Part  I.,  Kinematics 8vo,  1  50 

"  "  "  Part  II.,  Strength  and  Proportion  of 

Machine  Parts. 

Lanza's  Applied  Mechanics 8vo,  7  50 

MacCord's  Kinematics 8vo,  5  00 

Merriman's  Mechanics  of  Materials 8Vo,  4  00 

Metcalfe's  Cost  of  Manufactures 8vo,  5  00 

Michie's  Analytical  Mechanics Svo,  4  00 

Mosely's  Mechanical  Engineering.     (Mahari.) 8vo,  5  CO 

12 


Richards's  Compressed  Air 12mo,  $1  50 

Robinson's  Principles  of  Mechanism. 8vo,  3  00 

Smith's  Press- working  of  Metals 8vo,  i»  00 

The  Lathe  and  Its  Uses 8vo,  6  00 

Thurston's  Friction  and  Lost  Work 8vo,  3  00 

"        The  Animal  as  a  Machine 12mo,  1  00 

Warren's  Machine  Construction 2  vols.,  8vo,  7  50 

Weisbach's  Hydraulics  and  Hydraulic  Motors.    (Du  Bois.)..8vo,  5  00 
Mechanics    of   Engineering.      Vol.    III.,    Part   I., 

Sec.  I.     (Klein.) 8vo,  500 

Weisbach's   Mechanics    of  Engineering      Vol.    III.,    Part  I., 

Sec.  II      (Klein.) 8vo,  5  00 

Weisbach's  Steam  Engines.     (Du  Bois.) 8vo,  500 

Wood's  Analytical  Mechanics 8vo,  3  00 

' '      Elementary  Mechanics 12mo,  1  25 

"               "                  "           Supplement  and  Key 1  25 


METALLURGY. 

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Allen's  Tables  for  Iron  Analysis «.8vo,  3  00 

Egleston's  Gold  and  Mercury 8vo,  1  50 

"         Metallurgy  of  Silver 8vo,  750 

*  Kerl's  Metallurgy — Copper  and  Iron 8vo.  15  00 

*  "                            Steel.  Fuel,  etc 8vo,  1500 

Kuuhardt's  Ore  Dressing  in  Europe .8vo,  1  50 

Metcalf's  Si  eel— A  Manual  for  Steel  Users 12mo,  2  00 

O'Driscoll's  Treatment  of  Gold  Ores 8vo,  2  00 

Tiiurstcm's  Iron  and  Steel 8vo,  3  50 

Alloys 8vo,  250 

Wilson's  Cyanide  Processes 12mo,  1  50 

MINERALOGY   AND  MINING. 

MINE  ACCIDENTS — VENTILATION — ORE  DRESSING,  ETC. 

Barriuger's  Minerals  of  Commercial  Value oblong  morocco,  2  50 

Beard's  Ventilation  of  Mines 12mo,  2  50 

Boyd's  Resources  of  South  Western  Virginia 8vo,  3  00 

"      Map  of  South  Western  Virginia Pocket-book  form,  2  00 

Brush  and  Peufield's  Determinative  Mineralogy 8vo,  3  50 

Chester's  Catalogue  of  Minerals 8vo,  1  25 

"        "         paper,  50 

"        Dictionary  of  the  Names  of  Minerals 8vo,  3  00 

Dana's  American  Localities  of  Minerals 8vo,  1  00 

13 


Dana's  Descriptive  Mineralogy.     (E.  S.) 8vo,  half  morocco,  $12  50 

Mineralogy  and  Petrography    (J.D.) 12mo,  200 

"      Minerals  and  How  to  Study  Them.     (E.  S.) 12mo,  1  50 

"      Text-book  of  Mineralogy.     (E.  S.) 8vo,  3  50 

*Drinker's  Tunnelling,  Explosives,  Compounds,  and  Rock  Drills. 

4to,  half  morocco,  25  00 

Eglestou's  Catalogue  of  Minerals  and  Synonyms 8vo,  2  50 

Eissler's  Explosives — Nitroglycerine  and  Dynamite 8vo,  4  00 

Goodyear's  Coal  Mines  of  the  Western  Coast 12mo,  2  50 

Hussak's  Rock  forming  Minerals      (Smith.) 8vo,  2  00 

Ihlseng's  Manual  of  Mining   .    . .   8vo,  4  00 

Kuuhardt's  Ore  Dressing  in  Europe 8vo.  1  50 

O'Driscoll's  Treatment  of  Gold  Ores  .  . . . 8vo,  2  00 

Rosenbusch's    Microscopical    Physiography  of    Minerals    and 

Rocks-     (Iddings  ) 8vo.  500 

Sawyer's  Accidents  in  Mines 8vo,  7  00 

Stockbridge's  Rocks  and  Soils. . .8vo  2  50 

Walke's  Lectures  on  Explosives .8vo,  4  00 

Williarns's  Lithology 8vo,  3  00 

Wilson's  Mine  Ventilation I6mo,  125 

"        Hydraulic  and  Placer  Mining 12mo. 

STEAM  AND  ELECTRICAL  ENGINES,  BOILERS,  Etc. 

STATIONAKY — MARINE— LOCOMOTIVE — GAS  ENGINES,  ETC. 
(See  also  ENGINEERING,  p.  6.) 

Baldwin's  Steam  Heating  for  Buildings 12mo,  2  50 

Clerk's  Gas  Engine t 12mo;  4  00 

Ford's  Boiler  Making  for  Boiler  Makers 18mo,  1  00 

Hemen way's  Indicator  Practice 12mo.  2  00 

Hoadley's  Warm-blast  Furnace 8vo,  1  50 

Kneass's  Practice  and  Theory  of  the  Injector 8vo,  1  50 

MacCord's  Slide  Valve 8vo,  2  00 

*  Maw's  Marine  Engines Folio,  half  morocco,  18  00 

Meyer's  Modern  Locomotive  Construction 4to,  10  00 

Peabody  and  Miller's  Steam  Boilers 8vo,  4  00 

Peabody's  Tables  of  Saturated  Steam. , 8vo,  1  00 

"          Thermodynamics  of  the  Steam  Engine 8vo,  5  00 

Valve  Gears  for  the  Steam  Engine 8vo,  2  50 

Pray's  Twenty  Years  with  the  Indicator Royal  8vo,  2  50 

Pupin  and  Osterberg's  Thermodynamics 12mo,  1  25 

Reagan's  Steam  and  Electrical  Locomotives 12mo,  2  00 

Rontgen's  Thermodynamics.     (Du  Bois. ) 8vo,  5  00 

Sinclair's  Locomotive  Running 12mo,  2  00 

Thurston's  Boiler  Explosion 12mo,  1  50 

14 


Thurstou's  Engine  and  Boiler  Trials 8vo,  $5  00 

"  Manual  of  the  Steam  Engine.      Part  I.,  Structure 

and  Theory, 8vo,  7  50 

Manual  of  the   Steam  Eugine.      Part  II.,    Design, 

Construction,  and  Operation 8vo,  7  50 

2  parts,  12  00 

"           Philosophy  of  the  Steam  Engine 12mo,  75 

"  Reflection  on  the  Motive  Power  of  Heat.    (Caruot.) 

12mo,  1  50 

"           Stationary  Steam  Engines 12mo,  1  50 

"           Steam-boiler  Construction  and  Operation 8vo,  5  00 

Spaugler's  Valve  Gears 8vo,  2  50 

Trowbridge's  Stationary  Steam  Engines 4to,  boards,  2  50 

Weisbach's  Steam  Eugine.     (Du  Bois.) 8vo,  5  00 

Whithani's  Constructive  Steam  Engineering 8vo,  10  00 

' '           Steam-engine  Design r 8vo,  5  00 

"Wilson's  Steam  Boilers.     (Flather.) 12rno,  2  50 

Wood's  Thermodynamics,  Heat  Motors,  etc 8vo,  4  00 

TABLES,  WEIGHTS,  AND  MEASURES. 

FOR  ACTUARIES,  CHEMISTS,  ENGINEERS,  MECHANICS— METRIC 
TABLES,  ETC. 

Adriance's  Laboratory  Calculations 12mo,  1  25 

Allen's  Tables  for  Iron  Analysis.. 8vo,  3  00 

Bixby's  Graphical  Computing  Tables Sheet,  25 

Compton's  Logarithms 12mo,  1  50 

Crandall's  Railway  and  Earthwork  Tables 8vo,  1  50 

Egleston's  "Weights  and  Measures 18mo,  75 

Fisher's  Table  of  Cubic  Yards Cardboard,  25 

Hudson's  Excavation  Tables.     Vol.11 8vo,  100 

Johnson's  Stadia  and  Earthwork  Tables 8vo,  1  25 

Ludlow's  Logarithmic  and  Other  Tables.     (Bass.) 12mo,  2  00 

Thurston's  Conversion  Tables .  8vo,  1  00 

Totteu's  Metrology 8vo,  2  50 

VENTILATION. 

STEAM  HEATING — HOUSE  INSPECTION — MINE  VENTILATION. 

Baldwin's  Steam  Heating 12mo.  250 

Beard's  Ventilation  of  Mines 12mo,  2  50 

Carpenter's  Heating  and  Ventilating  of  Buildings 8vos  3  00 

Gerhard's  Sanitary  House  Inspection Square  16uio,  1  00 

Mott's  The  Air  We  Breathe,  and  Ventilation 16mo,  1  00 

Reid's  Ventilation  of  American  Dwellings 12mo,  1  50 

Wilson's  Mine  Ventilation , ItJmo,  1  25 

15 


MISCELLANEOUS  PUBLICATIONS. 

Alcott's  Gems,  Sentiment,  Language Gilt  edges,  $5  00 

Bailey's  The  New  Tale  of  a  Tub 8vo,  75 

Ballard's  Solution  of  the  Pyramid  Problem 8vo,  1  50 

Barnard's  The  Metrological  System  of  the  Great  Pyramid.  .8vo.  1  50 

Davis's  Elements  of  Law 8vo,  2  00 

Emmon's  Geological  Guide-book  of  the  Rocky  Mountains.  .8vo,  1  50 

Ferrel's  Treatise  on  the  Winds 8vo,  4  00 

Haines's  Addresses  Delivered  before' the  Am.  Ry.  Assn.  ..12mo.  2  50 

Mott's  The  Fallacy  of  the  Present  Theory  of  Sound.  .Sq.  16mo,  1  00 

Perkins's  Cornell  University Oblong  4to,  1  50 

Ricketts's  History  of  Rensselaer  Polytechnic  Institute 8vo,  3  00- 

Rotherham's    The    New    Testament     Critically    Emphasized. 

12m  o,  1  50 
The  Emphasized  Ifew  Test.     A  new  translation. 

Large  8vo,  2  00- 

Totteu's  An  Important  Question  in  Metrology 8vo,  2  50 

Whitehouse's  Lake  Moeris „ Paper,  25 

*  Wiley's  Yosemite,  Alaska,  and  Yellowstone 4to,  3  00 

HEBREW  AND   CHALDEE   TEXT=BOOKS. 

FOR  SCHOOLS  AND  THEOLOGICAL  SEMINARIES. 

Gesenius's  Hebrew  and    Chaldee   Lexicon   lo  Old    Testament. 

(Tregelles.) Small  4to,  half  morocco,  5  00 

Green's  Elementary  Hebrew  Grammar 12mo,  1  25 

Grammar  of  the  Hebrew  Language  (New  Edition). 8 vo,  3  00 

"       Hebrew  Chrestomathy 8vo,  2  00 

Letteris's    Hebrew   Bible   (Massoretic   Notes   in   English). 

8vo;  arabesque,  2  25 
Luzzato's  Grammar  of  the  Biblical  Chaldaic  Language  and   the 

Talmud  Babli  Idioms 12mo,  1  50 

MEDICAL. 

Bull's  Maternal  Management  in  Health  and  Disease 12mo,  1  00 

Haunmarsteu's  Physiological  Chemistry.    (Mandel.) 8vo,  4  00 

Mott's  Composition,  Digestibility,  and  Nutritive  Value  of  Food. 

Large  mounted  chart,  1  25 

Ruddiman's  Incompatibilities  in  Prescriptions 8vo,  2  00 

Steel's  Treatise  on  the  Diseases  of  the  Ox 8vo,  G  00 

Treatise  on  the  Diseases  of  the  Dog 8vo,  3  50 

Woodhull's  Military  Hygiene 12mo,  1  50 

Worcester's  Small  Hospitals— Establishment  and  Maintenance, 
including  Atkinson's  Suggestions  for  Hospital  Archi- 
tecture   12mo,  1  25 

16 


?fi 


U.C.  BERKELEY  LIBRARIES 


(100538^27 


